‘Social’ versus ‘Asocial’ cells--- Dynamic Competition Flux Balance Analysis

‘Social’ versus ‘Asocial’ cells--- Dynamic Competition Flux Balance Analysis | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article ‘Social’ versus ‘Asocial’ cells--- Dynamic Competition Flux Balance Analysis Yanhua Liu, Hans Westerhoff This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3059897/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 28 Oct, 2023 Read the published version in npj Systems Biology and Applications → Version 1 posted 10 You are reading this latest preprint version Abstract In multicellular organisms, different cell types compete for resources or growth factors, endangering cellular diversity as well as co-existence. To address this, we developed ‘dynamic cell-cell competition FBA’ (dcFBA). With total biomass synthesis as objective, we found that lower-growth-yield cell types face extinction even when they synthesized mutually required metabolic commodities. Signal transduction between cells promoted co-existence, when turning the cells into mutually regulatory and responsive ‘social cells’. Mutants with specific growth rate but intact signal transduction did not outgrow others. However, loss of its social characteristics enabled a mutant to dominate the other cell types with higher specific growth rates and bring those to extinction. A corollary is that cancer arises from reduced sensitivity to regulatory factors rather than enhanced specific growth rates. Therapies reinforcing cells’ cross-regulation, perhaps through alternative signaling routes, may therefore be more effective than those targeting replication rates. Biological sciences/Systems biology Biological sciences/Computational biology and bioinformatics Cell-cell interaction tumorigenesis systems biology flux balance analysis multicellularity Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Introduction The cells in a multicellular organism or stable ecosystem require regulation and coordination. Otherwise dysfunction and disease will develop over time. In the human body, cells from different organs have different inherent growth and turnover rates (e.g. peripheral T and B cells are renewed for 30%~40% every 48h 1 , red blood cells every 120 days 2 and brain cells rarely 3 ). Yet the effective proliferation rates of the different cell types should be virtually identical. The phenomenon of ‘Cell competition’ was first mentioned in a study of ‘ Minutes ’ mutants in Drosophila’s cell division rate 4 . The Minute cells grew more slowly than the normal cells, resulting in their elimination from the competition. Cell-cell competition for limited nutrients, growth factors, or space can optimize tissue fitness by eliminating ill-functioning cells through apoptosis 5 , but can also be exploited by super-competitor or tumor cells to kill neighboring normal cells. Cells with high Myc expression outcompete low Myc expressing cells 6 . Cell-cell competition involves metabolism, signaling pathways regulating growth, apoptosis, engulfment and interaction between winner and loser cells 7 , 8 . Super-competitor or tumor cells have alterations in two main pathways. Increased Myc gene expression suppresses survival signaling in neighbors (e.g. BMP-DPP signaling) 9 , 10 cells, while inactivation of their own Salvador-Warts-Hippo (Hippo) pathway affects cell growth, proliferation and apoptosis 11 . Resisting cell death is one of the hallmarks of tumor cells 12 which may be triggered by deactivation of the Hippo pathway. Paradoxically, the oncoprotein BCL2, correlates with a good short-term prognosis in breast cancer 13 , perhaps because it does not activate apoptosis in its neighbors. Tumor cells may activate their engulfment activity to induce apoptosis in surrounding cells 14 ; blocking engulfment enhances cell survival 15 . Normal epithelial cells have intrinsic anti-tumor activity 16 , inhibiting tumor progression. Avoiding immune destruction is another emerging hallmark of tumors 12 : competition between the immune system and tumor cells. In the normal situation, many transformed cells are destroyed by the immune system: Mice with both T cells’ and natural killer (NK) cells’ dysfunction have a higher probability of cancer development 17 , 18 . Tumor cells may escape from the immune system by secreting TGF-𝛽 or other immune-suppressive factors 19 , 20 . Additionally, tumor and host cells may compete for metabolic resources in their microenvironment 21 . The lactic acid secreted by many tumor cells, may inhibit neighboring cells or immune cells through acidification 22 , 23 , while ammonium secretion may do this for tumors with the WarburQ phenotype 24 . Understanding the mechanisms of cell competition and intercellular communication might help develop adjuvant therapies for diseases where the balances between different cell types are disturbed, such as imbalances between microbiome and body, autoimmune diseases, hyperplasia and cancer. An example of such an adjuvant therapy may be diets disadvantaging tumor cells exhibiting a Warburg effect 25 . Systems consisting of various cell types, replicating, undergoing apoptosis, competing for nutrients and communicating through growth factors rapidly become too complex to fathom by simple reasoning, however. In this paper we therefore examine whether systems biology could assist in understanding cell competition, including cells with oncogenic mutations. One way in which the different cell types of the human body are connected is metabolism: the cells compete for nutrition such as glucose, glutamine and oxygen, and together deliver mostly carbon dioxide and urea. Cells also help each other: lung cells help provide heart cells with oxygen and heart cells help lung cells by producing circulation, for instance. Averaged over hours, metabolism is at steady state, meaning that fluxes producing or importing any metabolite, balance fluxes degrading or exporting it. Flux Balance Analysis (FBA) 26 has become the method of choice for calculating flux balance in complex networks. To a system with multiple cell types competing for nutrition, standard FBA is not directly applicable however, as cell numbers and thereby metabolic fluxes vary over time. Dynamic FBA (dFBA) allows fluxes to vary with time at time scales longer than required for metabolic relaxation inside the cells. Consequently, for each metabolic intermediate synthesis plus import should continue to equal degradation plus export fluxes 27 , 28 . dFBA typically uses kinetic equations for dominant nutrient supply rates as functions of concentrations of growth substrates outside the network. However, it does not address the impact of cell number variation with time 29 . The cell concentrations within the network should be variable as well as dependent on the dynamic nutrient concentrations. Detailed kinetic modelling 30 , 31 should enable the modelling of metabolic networks with time variant cell numbers and nonlinearities of any type. However, kinetic modelling requires extensive kinetic details that are largely unknown for the topic at hand. Therefore, we here develop a method similar to FBA but capable of handling interacting and proliferating cell types with time-varying cell numbers. The new variant of FBA, which we shall refer to as dynamic cell-cell competition FBA (dcFBA) is applied to different cell types depending on each other through metabolites and/or growth factors. We find that such cross regulation needs to fulfill certain requirements in order to produce a stable organism consisting of the various cell types. Mutation to higher inherent growth rates should not in itself lead to tumorigenesis, but mutations in cross regulation should. Results 1. Towards dynamic competitive FBA 1–1 Neither competition for common substrate nor metabolic dependence on common goods produces stable coexistence in standard FBA with total biomass as objective In this paper we shall examine whether two (or three, see below) cell types that compete for a common metabolic substrate can reach steady coexistence. Figure 1 shows the metabolic network that we used in the flux balance analysis. It allows for the two cell types also to depend on each other through ‘common goods’ X (produced by cell type 1) and Y (produced by cell type 2) that both cell types require for their catabolism of glucose, their synthesis of X or Y and their growth (Fig. 1 A). Growth may have less than maximal yield because some glucose escapes to by-product; we call the corresponding flux rate ω. The scheme has 8 reactions, 5 metabolic intermediates (glucose, I1, I2, X, and Y) and one fixed flux (the glucose influx). A flux balance analysis (i.e. requiring steady state for the 5 intermediates) produced two modes of variation of the system. We selected the flux to by-product, and the difference between the biomass synthesis rates of cell type 2 and cell type 1 (which we called β 1 in Fig. 1 A and β in Fig. 1 B) as variables to monitor those modes of variation (Fig. 1 A). With total biomass synthesis as objective function, the flux to by-product drops to zero (Fig. 1 B). Sections 1 – 3 of the supplementary results show that the optimal flux always ran to the cheapest biomass, or to both cell types with different growth flux production. We conclude that competition for a common metabolic substrate and interdependence through common goods such as in Fig. 1 A, does not suffice to achieve coexistence of cell types. Figure 1 C and Fig. 1 D remind us that the actual network structure should be more complicated as it should also require balances around the concentrations of biomass 1 and biomass 2 (see also section 3 − 1 of Supplementary Results). These additions do not really affect to outcome of Fig. 1 B however (Fig. 1 D): One still obtains the situation that the more growth rate of the cheapest cell type becomes persistently higher than the specific growth rate of the more expensive cell type, so that no coexistence arises. Figure 1 C lacks one aspect of reality however. If the biomass synthesis flux of cell type 1 is lower than that of cell type 2, whilst the sum of the two fluxes must equal the maximum 0.5, the required synthesis rates of X and Y remain 1/h. The amount of biomass of type 1 will however decrease with time, so that, as time goes on, the same amount of X has to be synthesized by less and less of cell type 1. The standard FBA that we used up to this point does not accommodate the expectation that there should be a maximum to the X synthesis per unit biomass of cell type 1. When that maximum is achieved the growth cell type 2 should stop, perhaps causing coexistence. In the next sections we develop a variant of FBA that addresses these issues. 1–2 Preset exponential growth FBA for two cell types with capacity limitations We first effected of the tendency of the two cell types to grow exponentially by presetting exponential growth equations, as detailed in the Methods. As shown in Supplementary Fig. 9, if cell type 2 had a higher inherent specific growth rate, it always outgrew the cell type 1: still no stable coexistence developed. Paradoxically, as cell type 1 disappeared and with it the capacity to synthesize the common good X, cell type 2 continued to grow even though a shortage of X should arise. The required capacity for X synthesis, i.e. the flux per unit cell 1, increased to infinity (Supplementary Figs. 9D-9F). We then instated a maximum capacity by giving the X synthesis reaction a flux bound of three times the concentration of cell type 1. Figure 2 shows the consequences of this limitation of metabolic capacities for the preset growth model. Initially, Biomass 2 again increased whilst Biomass 1 decreased and with it the upper bound for X synthesis. After t = 12 h (for β(0) = 0.05, Fig. 2 A), the upper bound for X synthesis became lower than what was required, Biomass 2 abruptly decreased with time and the decrease with time of Biomass 1 accelerated. Total biomass thereby also decreased with time, ultimately to zero. For higher initial growth biases in favor of cell type 2, e.g. β(0) = 0.1 (Fig. 2 B) or β(0) = 0.2 (Fig. 2 C), the system behaved similarly, the transition happening earlier. The actual growth rate bias \(\left(t\right)=\frac{\text{b}2\left(\text{t}\right)-\text{b}1\left(\text{t}\right)}{2}\) changed with time to zero as indicated in Supplementary Fig. 10: No stable coexistence was reached in any of the cases studied. 1–3 Stepwise growth FBA for two cell types We also developed a ‘stepwise-growth FBA’ algorithm, whereby at each time point the rate of biomass synthesis is related to the cell number. The points in Supplementary Figs, 11A-C display the predicted cell numbers as function of time for three growth rate biases β. Again, one cell type ultimately outcompeted the other cell type, at any growth bias. We then incorporated maximum production capacities of X and Y per unit biomass of cell types 1 and 2 (as specified under Methods). Figure 3 demonstrates the outcome for three values of β(0). Again (cf. Figure 2 ) a greater growth bias difference led to a shorter lifespan for the system. Even when using the stepwise growth algorithm with maximum metabolic capacities of the two cell types, no stable coexistence was obtained. A kinetic model integrated by Copasi 32 did not reach coexistence either (Supplementary Fig. 12). 2. Regulation for preset-exponential growth and stepwise growth Here, we will explore whether explicit regulatory mechanisms might furnish co-existence of cell types differing in specific growth rates. 2 − 1 Preset-exponential growth FBA with capacity limits and cross regulation does not lead to stability either Now we introduce direct cross-regulation between the two cell types: The biomass synthesis rate of cell type 1 ( \({b}_{1,e,r ub}\left(t\right)\) ) will now be considered to be positively regulated (ε > 0) by the number of cells of type 2, as represented by a factor \({\left({B}_{2,e,r}\left(t\right)\right)}^{\epsilon }\) in the numerator (the negative dependence through the denominator continues to reflect the required Carbon flux balance, i.e. the competition for the glucose). The mechanism of this regulation might be a dependence of the expression level of an enzyme with much flux control on the specific growth rate of cell type 1. This dependence could involve gene expression regulation by a signal transduction route starting with a plasma membrane receptor addressed by a growth factor secreted by cell type 2 (see Methods). ε is then the ‘elasticity coefficient’ of the regulation 33 . We explored the effect of different regulatory strengths (i.e. ε = 0.5 or 1). Figure 4 shows that although this regulation postponed the time at which the system collapsed, it did not establish coexistence. 2–2 Stepwise growth FBA with cross regulation does show stability We then added the same type of cross regulation to the stepwise FBA algorithm with capacity limitation, again with an adjustable regulation elasticity \(\epsilon\) . Cell coexistence arose when the regulation power was strong enough (0.5 or 1; Fig. 5 ). The purple and yellow lines recall that in the absence of such regulation ( \(\epsilon =0\) ), no such stable coexistence developed. The minimum regulation power ( \(\epsilon\) value) for coexistence at three initial growth biases was 0.17 (β(0) = 0.05), 0.36 (β(0) = 0.1) and 0.92 (β(0) = 0.2). In the presence of the regulation, the steady state was reached before the maximum capacity was hit, so that the metabolic limitation was ineffective and even irrelevant for reaching coexistence. Accordingly, it was the cross regulation and not the limited metabolic capacities that made the two cell types co-exist. Complete regulation, i.e. an elasticity coefficient ε = 1, sufficed to bring the two cell types into coexistence even if their inherent growth rates differed by a factor of 9. At a high inherent growth bias, stable co-existence required a strong regulation power (Fig. 5 C). Also, in the kinetic model co-existence could be achieved when such regulation was put in (see Supplementary Fig. 14). 3. Two social cell types and an a-social mutant 3 − 1 An a-social mutant We then introduced a third cell type again under consideration of growth optimization for total biomass synthesis, flux balance, regulation and limited metabolic capacity (Fig. 6 ). We modeled the third cell type as a mutant of cell type 1 that continues to use common goods X and Y, that converts glucose to its intermediate I3 and that produces X, but has undergone a mutation interfering with the cross regulation with cell type 2. As a control we first considered a cell type 3 that was the same as type 1, i.e. neither autistic nor egoistic (Supplementary Fig. 16; ‘autistic’ and ‘egoistic’ mean, respectively, that the mutant is not responsive to regulation by cell type 2, or does not regulate cell type 2). The results in Supplementary Fig. 16 were the same as in Fig. 5 A with regulated power of 1. Then, there we considered three cases for the new mutant: 1) cell type 3 is both autistic and egoistic; 2) cell type 3 is autistic; 3) cell type 3 is egoistic. In case 1 , the mutant cell type 3 outgrew the other two cell types, bringing both cell numbers of the latter two to zero and its own numbers subsequently. The outgrowth of the mutant cells occurred even though it had the same (or lower) inherent specific growth rate (0.2) as cell type1 and even lower than that of cell type2 (Supplementary Fig. 17 and Fig. 7 ). In the ‘autism’ case 2, the mutant cell type 3 co-existed with cell type 2, but outgrew cell type 1 (Supplementary Fig. 18). Since the mutant (cell type 3) has the same functionality as its original (cell type 1), it did not break the system. In the ‘egoism’ case 3 the mutant could not outgrow the other two cell types and only existed at low concentrations, which may simulate a benign tumor (Supplementary Fig. 19). Next, we shall focus on case 1 for the three cell types system, with the stable situation of Fig. 5 A with \({\epsilon }=1\) as reference. Case 1 describes an ‘asocial cell type’ in a system comprising of social cell types, should outgrow the other cell types. This might seem remarkable because the signals it stopped responding to, appear to be stimulatory signals: an increase in the number of type 2 cells was modelled as enhancing the growth of the cell. After the mutation this stimulation was lost, suggesting that the mutant should be worse and not better off. However, the result is on the contrary, which is due to the growth regulating factor, which should be less than 1 to control excessive cell growth and should be proportional to cell numbers. 3 − 2 Three cell types regulating each other The emergence of the transformed cell type that has completely lost all regulation through mutation is unlikely. Most breast cancers are still estrogen dependent, for instance 34 , 35 . We therefore also considered the situation in which all three cell types continue to regulate each other at various elasticities. Specifically, we used the equations described in Methods to set the upper bound for biomass synthesis for each cell type and employed the stepwise growth dcFBA to determine the relationship between biomass synthesis, cell number and time. When we increased the reciprocal cross-regulation elasticity (𝛾 value) between the normal cells of type 1 and type 2 on the one hand and the ‘transformed’ cell type 3 on the other hand, we found that, already at the moderate regulation of 𝛾=0.5, cell type 3 no longer outgrew the other two cell types (Fig. 8 ). For an initial population composition of B 1 (0) = 0.39, B 2 (0) = 0.59 and B 3 (0) = 0.02, the minimal magnitude of 𝛾 for stability was around 0.3, at a death rate of 0.5/h. The cause of out-competing is linked to a lack of regulatory mechanisms in winner cells. Additionally, we investigated the higher growth rate of cell type 3 with and without regulation from other two cell types. As illustrated in Fig. S20, we found that cell type 3 is able to thrive at a high growth rate when regulation from the other two cell types is active, leading to the observed coexistence. Again, this result shows that cell regulation, rather than its inherent growth rate, is the determining factor for cells out-competing each other. 3–3 Only cell type 3 (mutant) being regulated by the other two cell types (normal cells) In actual situations, the transformed cell is still reliant on the normal cells’ activities, such as a tumor cell depending on the lung and the heart cell. However, the tumor cell may not support the normal cells in any way. When increasing the cross-regulation (𝛾 value) between the normal cells and the transformed cell, the transformed cell initially outgrew the other two cell types, and then this could subsequently lead to a coexistence of the three cell types (Fig. 9 C). However, this coexistence is not a typical occurrence since more often the normal cell count drastically decreases and approaches zero. As the regulation strength was increased to 0.4 or more, the two normal cell types outgrew the transformed cell (as shown in Fig. 9 D): if we increased the regulation power 𝛾 above 0.5. Discussion A new, ‘dynamic competitive FBA’ (dcFBA) methodology was developed to investigate the behavior of a multicellular system over time. Mathematical equations set the reaction bounds for a nutritionally competitive situation. Small time intervals were employed to iterate dynamic progress and flux balance requirement. For two cell types with different inherent growth rates, regulation was necessary to prevent one cell type from going extinct. In a three-cell type system involving two normal cell types and a transformed cell type, the loss of regulation led to the emergence of ‘asocial’ cells that exploited resources from others. We showed that this a-sociality and not a higher intrinsic growth rate was responsible for the mutant outgrowing the normal cells and endangering the persistence of the multicellular organism. The dcFBA developed here may also be useful for other issues around tissue homeostasis, such as the control of liver size. Within minutes after partial surgical hepatectomy, mammalian liver demonstrated regenerative abilities, recovering to its original organ mass within a matter of weeks 36 , 37 . The liver is also capable of returning to its normal size after hypertrophy and/or hyperplasia, by the activation of regression mechanisms (e.g. cell apoptosis), using mechanisms monitoring cell number or cell size 38 . Another example may reside in the competition between astrocytes and neurons, e.g. for amino acids uptake during brain development. In a previous paper 39 , we developed another mode of competitive FBA to demonstrate implications of the amino acid competition for uptake across the blood-brain barrier. One may now fruitfully combine the two new types of FBA and include neurotransmitter cross-talk and metabolism. FBA differs from kinetic modelling in requiring virtually no kinetic detail 26 . Since it is these kinetic details that are still missing for the complex networks at hand, this should make the dcFBA approach a useful addition to the modelling toolbox. FBA’s prediction of much fewer than all possible steady state behaviors, requires the assumption that network behavior is optimal however. The optimality is usually assumed to be maximal growth rate of the cells in question 26 and that is what we also assumed here: maximality of total biomass synthesis. Section 10 of the supplementary material shows that choosing the synthesis rate of either cell type 1 or cell type 2 as objective function, does not affect the essence of our conclusions. Our study did not deliver detail. The integration of more molecular, signaling and interaction information into FBA-kinetics hybrid models may lead to a more realistic representation of metabolic and signaling phenomena. MYC-mediated cell competition for instance is prevalent across different organs and tissues, including fibroblasts 40 and heart cells 41 , 42 , and plays a role in the growth and expansion of cancer cells 43 , 44 , 45 , 46 . Overexpression of MYC transforms tumor cells into super-competitors, enabling them to eliminate adjacent wild-type cells 47 . The growth signaling pathway in out-competed cells becomes impaired 48 , as evidenced by the decreased decapentaplegic transduction observed in out-competed cells 49 : the out-competed cells capture fewer growth factors. The winner cells increase their engulfment 14 , acquiring more space and resources while evading regulatory mechanisms from normal cells. Tumor cells also inactivate the Hippo pathway, thereby promoting their own proliferation 50 . The dcFBA developed here may be extended to deal with some of this complexity. Competition for nutrients, space, and growth factors is ubiquitous in multicellular organisms. On the one hand, normal or young cells compete with damaged or older cells to maintain tissue homeostasis; preventing such competition may result in lymphoblastic leukemia 51 . On the other hand, tumor or super-competitor cells become asocial when engaging in cell competition, leading to the elimination of normal cells. Our findings suggest that the critical time (i.e., the time before the tumor grows to a critical size) can be prolonged by increasing the initial cell number of normal cells (cell type 1 and type 2) (Supplementary Fig. 22). This may imply that an active body that has a low reduced glucose (metabolic substrate) level might delay the growth of tumor cells, suggesting an alternative to glucose fasting 52 . That also the cell-death rate constant ( \({k}_{D}\) ) affected the critical time (Supplementary Fig. 23), suggests that elderly individuals may be more sensitive to competition by tumor cells. Our results may further inspire the development of drugs or therapies targeting the competition or cross-regulation rather than cytotoxic agents or growth rate inhibitors: a higher inherent specific growth rate is unlikely to be the primary factor driving tumor cell growth (Fig. 7 and Supplementary Fig. 20). When social cells with a higher inherent specific growth rate were present, the asocial cell type (cell type 3) was able to outgrow them, but not when the latter was social and had a higher specific growth rate than other cell types. Our observations of the effects of cell-cell competition and cross-regulation may also be relevant more generally: Also, the components of an ecosystem are interdependent and single ‘dominant’ species or mutants rarely thrive in isolation. Relevant scenarios in global warming or human-induced pollution may be analyzed by our new dcFBA methodology with a proper objective function. Methods 1 Model building—Two cell types competing for common free-energy/carbon substrate and interacting through common goods. In all our models two (or three) cell types use glucose to produce their own metabolic intermediate (I1 and I2, respectively) (Supplementary Fig. 4). The so-called ‘common goods’ X and Y are produced by cell type 1 and 2, respectively, from their metabolic intermediate, and are both required by both cell types to produce their metabolic intermediate. Each cell type uses its intermediate either to produce its own biomass (i.e. to increase its cell number) or to produce one common good. The model also contains a by-product reaction from glucose and a carbon dioxide secreting reaction. In terms of Carbon, our models transduce half the glucose into biomass and the other half into carbon dioxide, a carbon/carbon yield that is not unusual 53 . The stoichiometries for the GTI1 reaction (see Supplementary Fig. 4) are taken to be -1 for glucose, X, and Y and + 1 for I1. For the GTI2 reaction they are analogously − 1 for glucose, X and Y and + 1 for I2. For DI1X they are − 1, + 4, and + 1 for I1, X, and CO 2 , respectively. For DI2Y they are − 1, + 4, and + 1 for I2, Y, and CO 2 , respectively. For the reaction towards biomass 1 they are − 1 and + 1 (or 1/b) for the intermediate I1 and the biomass 1, respectively. For synthesis of biomass 2, these numbers are − 1 and + 1 for the intermediate I2 and the biomass 2, respectively. In our simple model, only glucose was supplied in the medium. This model may correspond to the interaction between ‘lung cells’ and ‘liver cells’, which both depend for their growth on externally supplied glucose (for Carbon), glutamine (for nitrogen) and oxygen (for energetics) in the circulating blood. Lung cells oxygenate the blood, whilst liver cells enrich it with glutamine. Both cell types may use the glucose, glutamine and oxygen in growth and respiration producing carbon dioxide and urea. By using the COBRA routine 54 , 55 , FBA was performed with the sum of the two biomass production fluxes as objective, and the maximum total biomass synthesis flux was determined. This produced flux balances for glucose, X and Y, as well as I1 and I2. Acknowledging that COBRA’s FBA only yields a single flux pattern also when there are multiple equivalent patterns, we employed Flux Variability Analysis (FVA) 56 to compute the range of possible flux patterns for the same maximum magnitude of the objective function, by limiting each step to smaller absolute values. We also checked the effect of making biomass 1 ‘more expensive’ by making the production of biomass1 cost 2 units of I1, and biomass 2 ‘cheaper’ by requiring only one unit of I2 to produce 1 unit of biomass 2. In other models the biomass 1 synthesis reaction only produce 1/m C-moles of biomass per C-mole of I1. 2 Growth FBA—Dynamic cell competition Flux Balance Analysis for two cell types competing for a common substrate and with time varying cell densities, without or with capacity limitations, and with cross dependence through common goods X and Y According to the FBA, the biomass synthesis fluxes should optimally add up to 0.5 (This was because we assumed the glucose uptake equaled 1. In case where the glucose uptake differs from this value, the same equations should be applied with appropriate adjustment.). Assuming that this optimum is achieved at all times, i.e. also as the biomass concentration ratios change, the biomass synthesis fluxes for biomass 1 and biomass 2, respectively, should tend towards: $${b}_{1,e,ub}\left(t\right)=0.5\bullet \frac{{\mu }_{1}\bullet {f}_{1}\left(t\right)}{{\mu }_{1}\bullet {f}_{1}\left(t\right)+{\mu }_{2}{\bullet f}_{2}\left(t\right)}$$ 1 $${b}_{2,e,ub}\left(t\right)=0.5\bullet \frac{{\mu }_{2}\bullet {f}_{2}\left(t\right)}{{\mu }_{1}\bullet {f}_{1}\left(t\right)+{\mu }_{2}{\bullet f}_{2}\left(t\right)}$$ 2 with arbitrary functions f 1 (t) and f 2 (t) and \({\mu }_{1} \text{a}\text{n}\text{d} {\mu }_{2}\) representing the inherent specific growth rates. 2 − 1 Preset-exponential growth FBA for two cell types with cross dependence through common goods, obeying the optimal FBA condition of constant optimal total biomass flux As growth of cells tends to be exponential, we assume exponential functions of time for the two growth tendencies f(t) , i.e.: $${b}_{1,e,ub}\left(t\right)= 0.5\bullet \frac{{\mu }_{1}\bullet {e}^{\frac{{\mu }_{1}\bullet t}{{\mu }_{1}+{\mu }_{2}}}}{{\mu }_{1}\bullet {e}^{\frac{{\mu }_{1}\bullet t}{{\mu }_{1}+{\mu }_{2}}}+{\mu }_{2}\bullet {e}^{\frac{{\mu }_{2}\bullet t}{{\mu }_{1}+{\mu }_{2}}}}$$ 3 $${b}_{2,e,ub}\left(t\right)= 0.5\bullet \frac{{\mu }_{2}\bullet {e}^{\frac{{\mu }_{2}\bullet t}{{\mu }_{1}+{\mu }_{2}}}}{{\mu }_{1}\bullet {e}^{\frac{{\mu }_{1}\bullet t}{{\mu }_{1}+{\mu }_{2}}}+{\mu }_{2}\bullet {e}^{\frac{{\mu }_{2}\bullet t}{{\mu }_{1}+{\mu }_{2}}}}$$ 4 µ 1 = 0.25- \(\beta\) and µ 2 = 0.25+ \(\beta\) are the specific growth rates for cell types 1 and 2 and assumed to be constant. We used the ‘ \(\frac{{\mu }_{1}}{{\mu }_{1}+{\mu }_{2}}\) ’ and ‘ \(\frac{{\mu }_{2}}{{\mu }_{1}+{\mu }_{2}}\) ’ to normalize the growth rates for both cell types here. The parameter \(\beta\) is the initial growth rate bias in favor of cell type 2. Because the growth rate is set a priori as an exponential function of time we call this the ‘preset-exponential FBA’ procedure. A familiar way to make FBA produce maximal flux through a reaction is to give the corresponding reaction an upper bound equal to that flux and all other reactions higher upper bounds. We therefore set the upper bounds for the biomass synthesis reactions equal to the above expressions and then carried out FBA for Supplementary Fig. 4 at subsequent time points, separated from each other by ts (a small (infinitesimal) amount of time units) and with glucose efflux 1/ ts . This produced \(\text{b}\text{1}\text{,e}\text{,FBA}\left(t\right)\) and \(\text{b}\text{2,}\text{e,FBA}\left(t\right)\) , which were indeed identical to the upper bounds. Acknowledging that the cells should also be subject to death processes (for which we used a first order process with rate constant k D (we chose k D to equal 0.5)), we calculated the Biomass concentrations for the two cell types at each time point from: $${B}_{1,e}\left(t\right)={B}_{1,e}\left(t-ts\right)\bullet \left(1-ts\bullet {k}_{D}\right)+ts\bullet \text{b}\text{1}\text{,e,FBA}\left(t-ts\right)$$ 5 $${B}_{2,e}\left(t\right)={B}_{2,e}\left(t-ts\right)\bullet \left(1-ts\bullet {k}_{D}\right)+ts\bullet \text{b}\text{2,}\text{e,FBA}\left(t-ts\right)$$ 6 where ts = 0.1; B 1 (0) = 0.4 and B 2 (0) = 0.6 for β = 0.05; B 1 (0) = 0.3 and B 2 (0) = 0.7 for β = 0.1; B 1 (0) = 0.1 and B 2 (0) = 0.9 for β = 0.2 (These values will be the same for subsequent calculations unless specifically stated). We chose these values because we want the whole system to be stable before system broken (new biomass flux equaling the death flux). The difference between the cell numbers at the beginning is in accordance with their relative growth rates. 2–2 Stepwise growth FBA for two cell types We next developed an FBA growth algorithm in which the growth kinetics would not be pre-defined. At the glucose branch we consider fluxes towards I1, I2 and an overflow flux \({\omega }\) , the rates of which we assume to be all proportional to the glucose concentration level. The biomass synthesis ratio of cell types will be proportional to their cell number ( \(\frac{{b}_{1,s,ub}\left(t\right)}{{b}_{2,s,ub}\left(t\right)}=\frac{{\mu }_{1}\bullet {B}_{1,s}\left(t-ts\right)}{{\mu }_{2}\bullet {B}_{2,s}\left(t-ts\right)}\) ) described as differing in µ , but this could just as well reflect a difference in yield. With the common good balance met and the glucose influx fixed to 1, the two fluxes towards biomass amount to: $${b}_{1,s,ub}\left(t\right)=\frac{{0.5\bullet \mu }_{1}\bullet {B}_{1,s}\left(t-ts\right)\bullet \left(1-{\omega }\right)}{{\mu }_{1}\bullet {B}_{1,s}\left(t-ts\right)+{\mu }_{2}\bullet {B}_{2,s}\left(t-ts\right)}$$ 7 $${b}_{2,s,ub}\left(t\right)=\frac{{0.5\bullet \mu }_{2}\bullet {B}_{2,s}\left(t-ts\right)\bullet \left(1-{\omega }\right)}{{\mu }_{1}\bullet {B}_{1,s}\left(t-ts\right)+{\mu }_{2}\bullet {B}_{2,s}\left(t-ts\right)}$$ 8 Because the growth rates are reset at every time point of computation, we call this the ‘stepwise-growth FBA’ procedure. The amount of X required is equal to half of the sum of these two fluxes. Hence the fluxes from glucose to I1 and I2: $${v}_{1,g}\left(t\right)=\frac{1-{\omega }}{4}+\frac{{0.5\bullet \mu }_{1}\bullet {B}_{1,s}\left(t-ts\right)\bullet \left(1-{\omega }\right)}{{\mu }_{1}\bullet {B}_{1,s}\left(t-ts\right)+{\mu }_{2}\bullet {B}_{2,s}\left(t-ts\right)}$$ 9 $${v}_{2,g}\left(t\right)=\frac{1-{\omega }}{4}+\frac{{0.5\bullet \mu }_{2}\bullet {B}_{2,s}\left(t-ts\right)\bullet \left(1-{\omega }\right)}{{\mu }_{1}\bullet {B}_{1,s}\left(t-ts\right)+{\mu }_{2}\bullet {B}_{2,s}\left(t-ts\right)}$$ 10 Total biomass synthesis equals: $${b}_{1,s,ub}\left(t\right)+{b}_{2,s,ub}\left(t\right)=\frac{1-{\omega }}{2}$$ 11 When asking for maximal total biomass synthesis, and if the metabolic capacities are unlimited, \({\omega }\) becomes equal to zero and disappears from the equations. At every time point FBA was carried out for the metabolic network of Supplementary Fig. 4, with b 1,s,ub and b 2,s,ub as the upper bounds for the biomass synthesis for cell type 1 and 2, respectively: this produced the biomass synthesis fluxes \(\text{b}\text{1,}\text{s,FBA}\left(t\right)\) and \(\text{b}\text{2,}\text{s,FBA}\left(t\right)\) , which were effectively equal to the upper bounds. At each time step the cell concentrations were calculated as the same as above from with biomass synthesis fluxes and the death rates. 2–3 Limiting metabolic capacities The FBA calculations using the methods specified up to this point did not acknowledge any possible capacity limitation for producing common good X or Y that would arise due to lack of sufficient cells of type 1 or 2, respectively: the model continued to predict biomass synthesis of cell type 2 even when cell type 1, the sole producer of the X required by cell type 2 for the synthesis of its intermediate I2 and thereby for its growth, had been outgrown. To address this issue, we adjusted the upper bound for X and Y production so as to be proportional to the cell numbers of type 1 and type 2, respectively. Specifically, we set the maximum abilities to produce X or Y to three times their respective calculated cell numbers. After setting the reaction bound, we again ran the FBA at every time point to compute the growth rates of two cell types (i.e. \(\text{b}\text{1,}\text{s}\text{,FBA}\left(t\right)\) and \(\text{b}\text{2,}\text{s,FBA}\left(t\right)\) ). By the usual multiplication of the biomass production fluxes by the duration of the time step and by correction for cell death, we then calculated the predicted cell numbers for each cell type. In an alternative methodology, 1- \({\omega }\) in the above equations was replaced by: $$1-{\omega }\left(t\right)=minimum(1;{ 4\bullet {V}_{max}\bullet B}_{1}\left(t-ts\right); {4\bullet {V}_{max}\bullet B}_{2}\left(t-ts\right))$$ 12 We also developed a kinetic model based on irreversible mass action rate equations and used Copasi 32 to simulate the cells’ growth dynamics (See supplementary material). 3 Regulation 3 − 1 Regulation in preset-exponential growth FBA We also considered cases where the two cell types depend on each other for their growth also more directly than through the common goods X and Y. We assumed that cell type 2 produced a growth factor that stimulated growth of cell type 1 without being consumed by it, and that the concentration of that growth factor was proportional to the number of cells of type 2. The regulation is assumed to depend on the activation of a receptor by the binding of growth factor: G + R⇋RG (13) Where G is the growth factor produced by cell type 2 and R is the corresponding receptor in the plasma membrane of cell type 1. At binding equilibrium: RF= \(\frac{\left[\text{R}\text{G}\right]}{\left[\text{R}\right]+\left[\text{R}\text{G}\right]}=\frac{\left[\text{G}\right]}{\left[\text{G}\right]+{K}_{d}}\) (14) where the regulation factor (RF) is the fraction of receptor bound to growth factor G; K d is the dissociation equilibrium constant. If α is the ratio of the production rate constant to the first-order dilution rate constant of G, then: $$\text{G}={\alpha }\cdot {B}_{2}$$ 15 RF= \(\frac{{\alpha }\cdot {B}_{2}}{{\alpha }\cdot {B}_{2}+{K}_{d}}\) = \(\frac{{B}_{2}}{{B}_{2}+\frac{{K}_{d}}{{\alpha }}}\) (16) The regulation factor will always be smaller than 1 and depend on B 2 . We took a case where B 2 << \(\frac{{K}_{d}}{{\alpha }}\) and \({K}_{d}={\alpha }\) , so that RF equaled B 2 , i.e. for simplicity we used B 2 (cell 2 number) to represent the activity factor with different strengths. We ensured that B 2 ranged between 0 and 1, by adjusting the unit for cell numbers (i.e. to a billion). To simulate the regulation, we made the rate of synthesis of cell type 1 proportional to RF (and hence to the concentration of cell type 2) taken to the power of the ‘elasticity’ \(\epsilon\) : $${b}_{1,e,r ub}\left(t\right)= 0.5\bullet \frac{(0.25-)\bullet {e}^{\frac{\left(0.25-\right)\bullet t}{0.5}} \bullet {\left({B}_{2,e,r}\left(t\right)\right)}^{\epsilon }}{(0.25-){\bullet e}^{\frac{\left(0.25-\right)\bullet t}{0.5}}\bullet {\left({B}_{2,e,r}\left(t\right)\right)}^{\epsilon }+(0.25+)\bullet {e}^{\frac{\left(0.25+\right)\bullet t}{0.5}}\bullet {\left({B}_{1,e,r}\left(t\right)\right)}^{\epsilon }}$$ 17 \(\epsilon\) is the elasticity coefficient of the regulation 33 akin to the power in Biochemical Systems Theory 57 . The corresponding dependence of synthesis of cell type 2 on the concentration of cell type 1 reads as follows: $${b}_{2,e,r,ub}\left(t\right)= 0.5\bullet \frac{(0.25+)\bullet {e}^{\frac{\left(0.25+\right)\bullet t}{0.5}}\bullet {\left({B}_{1,e,r}\left(t\right)\right)}^{\epsilon }}{(0.25-){\bullet e}^{\frac{\left(0.25-\right)\bullet t}{0.5}}\bullet {\left({B}_{2,e,r}\left(t\right)\right)}^{\epsilon }+(0.25+)\bullet {e}^{\frac{\left(0.25+\right)\bullet t}{0.5}}\bullet {\left({B}_{1,e,r}\left(t\right)\right)}^{\epsilon }}$$ 18 As before we used these equations to set the upper bounds for the biomass synthesis reactions in the metabolic scheme and then carried out FBA with the total biomass synthesis as objective. The resulting biomass synthesis for both cell types were again used to calculate the cell levels for the next time point as same as above. As the equation contains a predefined exponential growth tendency, we call this procedure ‘regulated preset-exponential growth FBA’. 3 − 2 Regulation in stepwise growth FBA We also introduced cross regulation between cell types, at a strength again indicated by the parameter ‘ \(\epsilon\) ’ in the ‘stepwise growth’ FBA model: $${b}_{1,s,r,ub}\left(t\right)=\frac{0.5{\bullet \mu }_{1}\bullet \left({B}_{1,s,r}\left(t-ts\right)\right)\bullet {\left({B}_{2,s,r}\left(t-ts\right)\right)}^{\epsilon }}{{\mu }_{1}\bullet \left({B}_{1,s,r}\left(t-ts\right)\right)\bullet {\left({B}_{2,s,r}\left(t-ts\right)\right)}^{\epsilon }+{\mu }_{2}\bullet {\left({B}_{1,s,r}\left(t-ts\right)\right)}^{\epsilon }\bullet \left({B}_{2,s,r}\left(t-ts\right)\right)}$$ 19 $${b}_{2,s,r,ub}\left(t\right)=\frac{0.5{\bullet \mu }_{2}\bullet {\left({B}_{1,s,r}\left(t-ts\right)\right)}^{\epsilon }\bullet \left({B}_{2,s,r}\left(t-ts\right)\right)}{{\mu }_{1}\bullet \left({B}_{1,s,r}\left(t-ts\right)\right)\bullet {\left({B}_{2,s,r}\left(t-ts\right)\right)}^{\epsilon }+{\mu }_{2}\bullet {\left({B}_{1,s,r}\left(t-ts\right)\right)}^{\epsilon }\bullet \left({B}_{2,s,r}\left(t-ts\right)\right)}$$ 20 Here b 1,s,r,ub and b 2,s,r,ub are again used to set the upper bounds for the biomass synthesis reactions. After that, we get actual biomass synthesis rates for cell type 1 and 2 respectively with regulation in stepwise growth. And we use these values to calculate the cell number for cell type 1 and 2 like above. Although the inherent specific growth rates for cell types 1 and 2 are µ 1 = 0.25- \(\beta\) and µ 2 = 0.25+ \(\beta\) in the beginning, respectively, the two actual specific growth rate tendencies may depend on time. We tried various regulation powers (i.e., different values of \(\epsilon\) ) to set the reaction upper bound for biomass production and to check how the cell number was changing with time. As the equation does not contain a predefined exponential growth tendency for the two cells types but this growth tendency is set by the stepwise change in biomass concentrations with time, we call this procedure ‘regulated stepwise growth FBA’. 4. Three cell types 4 − 1 Stepwise growth FBA for two social cell types together with one asocial cell type After calculating the system with two cell types, we added another cell type to the system. The third cell type was a mutant of cell type 1 deficient in the communication with cell type 2. Otherwise it functioned identically to cell type 1: It still had the ability to produce common good X, and used common goods X and Y whilst converting glucose to its intermediate metabolite I3. The model building file is provided in the folder ‘files’ in the GitHub directory. The influx was again 1 C-mole/h, the sum of biomass 1, biomass 2 and biomass 3 synthesis was again 0.5 C-mole/h. The excess carbon produced in the reactions forming X and Y was supposed to leave the system as CO 2 . We wanted to examine whether there could be coexistence of the three cell types. We first just studied the control case (i.e. \(\partial\) =1 and \(\epsilon 1=\epsilon ,\) equations below) in which cell type 3 was the same as cell type 1, i.e. with regulation to cell type 2 and responsive to cell type 2 regulation. Then, we considered three ‘asocial’ cases, i.e. case 1 (‘autistic and egoistic’, i.e. cell type 3 neither regulating/stimulating cell type 2, nor responsive to cell type 2 regulation, i.e. \(\partial\) =0 and \(\epsilon 1=0\) ), case 2 (‘autistic’, i.e. cell type 3 not regulated by cell type 2, cell type 2 still regulated by cell type 3), i.e., \(\partial\) =1 and \(\epsilon 1=0\) , respectively. We also considered the remaining case 3 (‘egoistic’, i.e. cell type 3 not regulating cell type 2, but still regulated by cell type 2, i.e. \(\partial\) =0 and \(\epsilon 1=\epsilon\) ). With the common goods balance met and the glucose influx fixed to 1, the three maximal fluxes towards biomass amounted to: Here ‘ \(\epsilon\) ’ or ‘ \(\epsilon 1\) ’ is the inter-regulation between cell type 1 or cell type 3 with type 2 in this three-cell-types’ stepwise growth. Unless specified otherwise, the inherent specific growth rate for cell types 1 and 3 were the same ( \({\mu }_{1}={\mu }_{3}=0.2\) ), while that of cell type 2 was higher ( \({\mu }_{2}=0.3\) ). As per the numerators of the above equations, we tried various values of \(\epsilon\) and used the above equations to set the upper bound for the biomass synthesis rate for the three cell types. The sum of the biomass synthesis rates of the three cell types was again the objective function. After defining the model, we calculated the relationship between the cell number and time by using COBRA at every time step and then integrating over time by the above equations for the Biomasses. Finally, we selected case 1 (both autistic and egoistic) for further analysis because that case cell type 3 was most similar in behavior to tumor cells. 4 − 2 Three cell types regulating each other We could not find a coexistence for three cell types when we chose various \(\epsilon\) values, even though cell type 3 needed support from the other two in terms of common goods X and Y. As cell type 3 modelled a tumor cell, we considered it unlikely that it had completely lost all regulation. Therefore, we next considered the regulatory interactions among three cell types, whereby cell types 1 and 2 mutually regulate each other with strength (elasticity 33 , 57 ) ε, and both cross-regulate with cell type 3, and vice versa , with a regulation strength (elasticity) 𝛾. The regulated equation is shown below. $${b}_{1,s3,r1,ub}\left(t\right)=\frac{{0.5\bullet \mu }_{1}\bullet {{B}_{1,s3,r1}\left(t-ts\right)\bullet \left({B}_{2,s3,r1}\left(t-ts\right)\right)}^{\epsilon }{\bullet \left({B}_{3,s3,r1}\left(t-ts\right)\right)}^{\gamma }}{FR1}$$ 26 $${b}_{2,s3,r1,ub}\left(t\right)=\frac{{0.5\bullet \mu }_{2}\bullet {\left({B}_{1,s3,r1}\left(t-ts\right)\right)}^{\epsilon }{\bullet {B}_{2,s3,r1}\left(t-ts\right)\bullet \left({B}_{3,s3,r1}\left(t-ts\right)\right)}^{\gamma }}{FR1}$$ 27 $${b}_{3,s3,r1,ub}\left(t\right)=\frac{0.5\bullet {\mu }_{3}{\bullet \left({B}_{1,s3,r1}\left(t-ts\right)\right)}^{\gamma }{\bullet \left({B}_{2,s3,r1}\left(t-ts\right)\right)}^{\gamma }\bullet {B}_{3,s3,r1}\left(t-ts\right)}{FR1}$$ 28 where ‘ \(\gamma\) ’ is the regulation power (elasticity) between the normal cells and cell type 3, and vice versa . \({b}_{1,s3,r1,ub}\) , \({b}_{2,s3,r1,ub}\) and \({b}_{3,s3,r1,ub}\) are again used to set the upper bounds for the biomass synthesis reactions. After that, we get the actual biomass synthesis rate for cell types 1, 2 and 3 in this three-cell-types system with regulation by dcFBA for each time point with the sum of their biomass synthesis rates as objective function, then these values will be used for cell number calculation. 4 − 3 Only cell type 3 (transformed cell) regulated by the two other cell types (normal cells) In actual situations, the transformed cell needs support from normal cells, but itself may not support the normal cells in any way. Accordingly, we constructed another regulation network through equations in which the normal cells regulated the transformed cells, and not vice versa : $$FR2={\mu }_{1}\bullet {{B}_{1,s3,r1}\left(t-ts\right)\bullet \left({B}_{2,s3,r1}\left(t-ts\right)\right)}^{\epsilon }+{\mu }_{2}\bullet {\left({B}_{1,s3,r1}\left(t-ts\right)\right)}^{\epsilon }\bullet {B}_{2,s3,r1}\left(t-ts\right)$$ $$+ {\mu }_{3}{\bullet \left({B}_{1,s3,r1}\left(t-ts\right)\right)}^{\gamma }{\bullet \left({B}_{2,s3,r1}\left(t-ts\right)\right)}^{\gamma }\bullet {B}_{3,s3,r1}\left(t-ts\right)$$ 29 $${b}_{1,s3,r2,ub}\left(t\right)=\frac{{0.5\bullet \mu }_{1}\bullet {{B}_{1,s3,r1}\left(t-ts\right)\bullet \left({B}_{2,s3,r1}\left(t-ts\right)\right)}^{\epsilon }}{FR2}$$ 30 $${b}_{2,s3,r2,ub}\left(t\right)=\frac{{0.5\bullet \mu }_{2}\bullet {\left({B}_{1,s3,r1}\left(t-ts\right)\right)}^{\epsilon }\bullet {B}_{2,s3,r1}\left(t-ts\right)}{FR2}$$ 31 $${b}_{3,s3,r2,ub}\left(t\right)=\frac{{0.5\bullet \mu }_{3}{\bullet \left({B}_{1,s3,r1}\left(t-ts\right)\right)}^{\gamma }{\bullet \left({B}_{2,s3,r1}\left(t-ts\right)\right)}^{\gamma }\bullet {B}_{3,s3,r1}\left(t-ts\right)}{FR2}$$ 32 Again, we used equation above to set the upper bound for the biomass production reaction and performed the dcFBA calculation with the sum of their biomass synthesis rates of three cell types as objective function. And we used the biomass synthesis value from dcFBA to calculate the cell number like above. Declarations DATA AVAILABILITY All data used for the simulations are presented in this manuscript and its Supplementary files. CODE AVAILABILITY The code of simulated models is available at Github: https://github.com/YanhuaLiu1/Cells-Competiition Author contributions Yanhua Liu devised the study, collected literature data, prepared the models, performed all computations, showed and discussed the results and drafted the manuscript. Both authors critically reviewed the data and the manuscript. Hans V. Westerhoff devised the study, discussed the results and edited the manuscript. All authors read and approved the final manuscript. Acknowledgements This study was supported by a personal development grant from the Guangzhou Elite Project (GEP) Foundation to Yanhua Liu for which we express our gratitude. 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Development (Cambridge, England) , 146 (3), dev170753. https://doi.org/10.1242/dev.170753 (2019). Xiao, G., et al. Inducible activation of c-Myc in adult myocardium in vivo provokes cardiac myocyte hypertrophy and reactivation of DNA synthesis. Circulation research , 89 (12), 1122–1129. https://doi.org/10.1161/hh2401.100742 (2001). Vita, M., & Henriksson, M. The Myc oncoprotein as a therapeutic target for human cancer. Seminars in cancer biology , 16 (4), 318–330. https://doi.org/10.1016/j.semcancer.2006.07.015 (2006). Di Giacomo, S., Sollazzo, M., Paglia, S., & Grifoni, D. MYC, Cell Competition, and Cell Death in Cancer: The Inseparable Triad. Genes , 8 (4), 120. https://doi.org/10.3390/genes8040120 (2017). Di Giacomo, S., et al. Human Cancer Cells Signal Their Competitive Fitness Through MYC Activity. Scientific reports , 7 (1), 12568. https://doi.org/10.1038/s41598-017-13002-1 (2017). Paglia, S., Sollazzo, M., Di Giacomo, S., Strocchi, S., & Grifoni, D. Exploring MYC relevance to cancer biology from the perspective of cell competition. Seminars in cancer biology , 63 , 49–59. https://doi.org/10.1016/j.semcancer.2019.05.009 (2020). Froldi, F., et al. The lethal giant larvae tumour suppressor mutation requires dMyc oncoprotein to promote clonal malignancy. BMC biology , 8 , 33. https://doi.org/10.1186/1741-7007-8-33 (2010). Moreno, E., Basler, K., & Morata, G. Cells compete for decapentaplegic survival factor to prevent apoptosis in Drosophila wing development. Nature , 416 (6882), 755–759. https://doi.org/10.1038/416755a (2002). Moreno, E., & Basler, K. dMyc transforms cells into super-competitors. Cell , 117 (1), 117–129. https://doi.org/10.1016/s0092-8674(04)00262-4 (2004). Han Y. Analysis of the role of the Hippo pathway in cancer. Journal of translational medicine , 17 (1), 116. https://doi.org/10.1186/s12967-019-1869-4 (2019). Martins, V., et al . Cell competition is a tumour suppressor mechanism in the thymus. 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Additional Declarations (Not answered) Supplementary Files supplementarymaterialswithfiguresandtables.docx Cite Share Download PDF Status: Published Journal Publication published 28 Oct, 2023 Read the published version in npj Systems Biology and Applications → Version 1 posted Editorial decision: revise 14 Jul, 2023 Review # 2 received at journal 13 Jul, 2023 Review # 1 received at journal 03 Jul, 2023 Reviewer # 2 agreed at journal 21 Jun, 2023 Reviewer # 1 agreed at journal 19 Jun, 2023 Reviewers invited by journal 18 Jun, 2023 Submission checks completed at journal 15 Jun, 2023 First submitted to journal 14 Jun, 2023 Unknown event 14 Jun, 2023 Editor assigned by journal 13 Jun, 2023 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-3059897","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":210977081,"identity":"5d596a7b-d5c8-43ec-99f4-651139ec483b","order_by":0,"name":"Yanhua Liu","email":"","orcid":"https://orcid.org/0000-0001-8282-7404","institution":"Swammerdam Institute for Life Sciences, University of Amsterdam","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Yanhua","middleName":"","lastName":"Liu","suffix":""},{"id":210977082,"identity":"e8b02b4f-7317-43f0-bc15-bc7fc17120a6","order_by":1,"name":"Hans Westerhoff","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA6UlEQVRIiWNgGAWjYBACAwSD8cGHD0AGGztRWhJADGbDmTNAWphJ0TKbB8QhpMVcIsfw4c8fNvnm7M2MzTa/tsnzMTMwfviYg1uL5YwcY2OehDTLnT2HGZtz+24btjEzMEvO3IbHYWfObpNmSDhsYHAj//jj3J7bjEAtbMy8+LVs//kj4T9QSzJjs2XPbXvCWo73bmPgSTgA0cLw43YiEVr6P0vzpCUbGJw5zNjY23A7uY2ZsRm/Xw6zJX78YWNnYHC8mbHhx5/btvPbmw9++IhHCypgbAOTDcSqB4E/pCgeBaNgFIyCkQIAxmhTRL8Q0ocAAAAASUVORK5CYII=","orcid":"","institution":"Swammerdam Institute for Life Sciences, University of Amsterdam","correspondingAuthor":true,"submittingAuthor":false,"prefix":"","firstName":"Hans","middleName":"","lastName":"Westerhoff","suffix":""}],"badges":[],"createdAt":"2023-06-13 21:13:39","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3059897/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3059897/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1038/s41540-023-00313-5","type":"published","date":"2023-10-28T04:00:00+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":39107538,"identity":"c7a73e24-4092-49dc-8cc4-5aeb380b07ef","added_by":"auto","created_at":"2023-06-26 18:28:28","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":92433,"visible":true,"origin":"","legend":"\u003cp\u003eSee image above for figure legend.\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-3059897/v1/8864eef900450142f7e75875.png"},{"id":39107539,"identity":"b99e2172-b666-43ad-8daa-a6086fe2d6b8","added_by":"auto","created_at":"2023-06-26 18:28:28","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":148110,"visible":true,"origin":"","legend":"\u003cp\u003eBiomass levels (B1, B2) and related biomass synthesis rates (b1 and b2) and the required X and Y production rates for cell type 1 and cell types 2, calculated for different specific growth bias values (0.05 (A\u0026amp;D), 0.1 (B\u0026amp;E), 0.2 (C\u0026amp;F)) and with capacities of metabolic reactions limited through ‘\u003cem\u003eexponential\u003c/em\u003e’ competition as described under Materials and Methods. Glucose influx was unlimited and glucose efflux absent.\u0026nbsp; The FBA objective was maximal total biomass synthesis.\u0026nbsp; These were calculated by adding the balance between b1 and b2 and the death rate (Biomass 1 or Biomass 2 respectively, multiplied by \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003eD\u003c/em\u003e\u003c/sub\u003e=0.5) after every time step of 0.1 h.\u0026nbsp; A total specific biomass synthesis of 0.5 requires an X synthesis flux of 0.25.\u0026nbsp; The maximum synthesis flux of X catalyzed by cell type 1 was taken to equal 3 per unit Biomass 1.\u0026nbsp; Hence at a cell number of 0.25/3=0.083 the amount of B1 stopped to suffice for the synthesis of the X required for the maximum total biomass synthesis of 0.5, leading to the drops in the optimal b1 and b2 predicted by the FBA.\u0026nbsp; It was checked that the FBA solutions obtained corresponded to flux balances throughout the metabolic network. The cell number scale is arbitrary but could be taken as 1 billion.\u0026nbsp; Also, the time scale is arbitrary, shown here as hour but may realistically be closer to 1 month.\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-3059897/v1/b63972b7e71537269c890958.png"},{"id":39107546,"identity":"18088412-6032-4605-b200-9abc04549673","added_by":"auto","created_at":"2023-06-26 18:28:29","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":145841,"visible":true,"origin":"","legend":"\u003cp\u003eBiomass levels (B1, B2) and related biomass synthesis rate (b1 and b2) and the required X and Y production for cell type 1 and cell types 2, respectively, calculated for different specific growth bias values (0.05 (A\u0026amp;D), 0.1 (B\u0026amp;E), 0.2 (C\u0026amp;F)) and with capacities of metabolic reactions (I1 to X and I2 to Y) limited to 3 times the corresponding Biomass level, as calculated using the stepwise-growth FBA algorithm as described under Materials and Methods. Glucose influx was unlimited and glucose efflux absent.\u0026nbsp; The FBA objective was maximal total biomass synthesis, calculated by adding the balance between b1 and b2 and the death rate (Biomass 1 or Biomass 2 respectively, multiplied by \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003eD\u003c/em\u003e\u003c/sub\u003e=0.5) after every time step of 0.1 month.\u0026nbsp;\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-3059897/v1/e1ff31e7ad6db0de718e6023.png"},{"id":39110378,"identity":"e2f932fb-12a8-4cb7-a0b8-a514e4e9014d","added_by":"auto","created_at":"2023-06-26 18:44:28","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":180390,"visible":true,"origin":"","legend":"\u003cp\u003eSee image above for figure legend.\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-3059897/v1/642a77841154f0ae180426c7.png"},{"id":39110376,"identity":"2effec8f-5983-4316-9b92-584c2c838319","added_by":"auto","created_at":"2023-06-26 18:44:28","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":154216,"visible":true,"origin":"","legend":"\u003cp\u003eSee image above for figure legend.\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-3059897/v1/975c865b52ccce35a57c1d80.png"},{"id":39107540,"identity":"5a8316d0-7485-4dd9-bc2e-027810dae1ff","added_by":"auto","created_at":"2023-06-26 18:28:28","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":97028,"visible":true,"origin":"","legend":"\u003cp\u003eThe three cell types’ networks. Cell type 3 is a mutant of cell type 1 just losing the regulation by and of cell type 2, and is taken to exemplify an ‘asocial cell’ as it does not engage in cross regulation. \u0026nbsp;All three cell types equally require X and Y for growth. Cell types 1 and 3 both produce X whilst cell type 2 produces Y.\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-3059897/v1/d0a789e981504ecec7980335.png"},{"id":39107543,"identity":"a33b0b18-e059-42c7-bbfb-52535cc84c56","added_by":"auto","created_at":"2023-06-26 18:28:28","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":145186,"visible":true,"origin":"","legend":"\u003cp\u003eSee image above for figure legend.\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-3059897/v1/15b5eac947fb1f9a29976c7c.png"},{"id":39109122,"identity":"13cf3790-6683-4174-bb65-a2d55df2f34a","added_by":"auto","created_at":"2023-06-26 18:36:28","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":114062,"visible":true,"origin":"","legend":"\u003cp\u003eSee image above for figure legend.\u003c/p\u003e","description":"","filename":"8.png","url":"https://assets-eu.researchsquare.com/files/rs-3059897/v1/033da1a09478f5fc632e33f1.png"},{"id":39110377,"identity":"072a98ba-3253-432a-8173-6415c7f35c66","added_by":"auto","created_at":"2023-06-26 18:44:28","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":111198,"visible":true,"origin":"","legend":"\u003cp\u003eSee image above for figure legend.\u003c/p\u003e","description":"","filename":"9.png","url":"https://assets-eu.researchsquare.com/files/rs-3059897/v1/3cf545b77fe7d5f8b4a00008.png"},{"id":45394797,"identity":"aefc6353-f5ac-4659-aa85-e4e7f07cedd2","added_by":"auto","created_at":"2023-10-29 07:13:45","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1600007,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3059897/v1/910d4604-c4c2-4dd0-93e0-80ba264032ee.pdf"},{"id":39107547,"identity":"26b48f93-dcb1-4ab3-a0fb-e7d7a6d75545","added_by":"auto","created_at":"2023-06-26 18:28:29","extension":"docx","order_by":2,"title":"","display":"","copyAsset":false,"role":"supplement","size":11605354,"visible":true,"origin":"","legend":"","description":"","filename":"supplementarymaterialswithfiguresandtables.docx","url":"https://assets-eu.researchsquare.com/files/rs-3059897/v1/277a6247446b044ff763c6c6.docx"}],"financialInterests":"(Not answered)","formattedTitle":"‘Social’ versus ‘Asocial’ cells--- Dynamic Competition Flux Balance Analysis","fulltext":[{"header":"Introduction","content":"\u003cp\u003eThe cells in a multicellular organism or stable ecosystem require regulation and coordination. Otherwise dysfunction and disease will develop over time. In the human body, cells from different organs have different inherent growth and turnover rates (e.g. peripheral T and B cells are renewed for 30%~40% every 48h\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e1\u003c/span\u003e\u003c/sup\u003e, red blood cells every 120 days\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e and brain cells rarely\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e). Yet the \u003cem\u003eeffective\u003c/em\u003e proliferation rates of the different cell types should be virtually identical.\u003c/p\u003e \u003cp\u003eThe phenomenon of \u0026lsquo;Cell competition\u0026rsquo; was first mentioned in a study of \u0026lsquo;\u003cem\u003eMinutes\u003c/em\u003e\u0026rsquo; mutants in Drosophila\u0026rsquo;s cell division rate\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e. The \u003cem\u003eMinute\u003c/em\u003e cells grew more slowly than the normal cells, resulting in their elimination from the competition. Cell-cell competition for limited nutrients, growth factors, or space can optimize tissue fitness by eliminating ill-functioning cells through apoptosis\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e, but can also be exploited by super-competitor or tumor cells to kill neighboring normal cells. Cells with high Myc expression outcompete low Myc expressing cells\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eCell-cell competition involves metabolism, signaling pathways regulating growth, apoptosis, engulfment and interaction between winner and loser cells\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e\u003c/span\u003e,\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e. Super-competitor or tumor cells have alterations in two main pathways. Increased Myc gene expression suppresses survival signaling in neighbors (e.g. BMP-DPP signaling)\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e\u003c/span\u003e,\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e cells, while inactivation of their own Salvador-Warts-Hippo (Hippo) pathway affects cell growth, proliferation and apoptosis\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e. Resisting cell death is one of the hallmarks of tumor cells\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e which may be triggered by deactivation of the Hippo pathway. Paradoxically, the oncoprotein BCL2, correlates with a good short-term prognosis in breast cancer\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e, perhaps because it does not activate apoptosis in its neighbors. Tumor cells may activate their engulfment activity to induce apoptosis in surrounding cells\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e; blocking engulfment enhances cell survival\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eNormal epithelial cells have intrinsic anti-tumor activity\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e, inhibiting tumor progression. Avoiding immune destruction is another emerging hallmark of tumors\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e: competition between the immune system and tumor cells. In the normal situation, many transformed cells are destroyed by the immune system: Mice with both T cells\u0026rsquo; and natural killer (NK) cells\u0026rsquo; dysfunction have a higher probability of cancer development\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e\u003c/span\u003e,\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e. Tumor cells may escape from the immune system by secreting TGF-\u0026#120573; or other immune-suppressive factors\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e\u003c/span\u003e,\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e. Additionally, tumor and host cells may compete for metabolic resources in their microenvironment\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e. The lactic acid secreted by many tumor cells, may inhibit neighboring cells or immune cells through acidification\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e\u003c/span\u003e,\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e, while ammonium secretion may do this for tumors with the WarburQ phenotype\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eUnderstanding the mechanisms of cell competition and intercellular communication might help develop adjuvant therapies for diseases where the balances between different cell types are disturbed, such as imbalances between microbiome and body, autoimmune diseases, hyperplasia and cancer. An example of such an adjuvant therapy may be diets disadvantaging tumor cells exhibiting a Warburg effect\u003csup\u003e\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e\u003c/sup\u003e. Systems consisting of various cell types, replicating, undergoing apoptosis, competing for nutrients and communicating through growth factors rapidly become too complex to fathom by simple reasoning, however. In this paper we therefore examine whether systems biology could assist in understanding cell competition, including cells with oncogenic mutations.\u003c/p\u003e \u003cp\u003eOne way in which the different cell types of the human body are connected is metabolism: the cells compete for nutrition such as glucose, glutamine and oxygen, and together deliver mostly carbon dioxide and urea. Cells also help each other: lung cells help provide heart cells with oxygen and heart cells help lung cells by producing circulation, for instance. Averaged over hours, metabolism is at steady state, meaning that fluxes producing or importing any metabolite, balance fluxes degrading or exporting it. Flux Balance Analysis (FBA)\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e has become the method of choice for calculating flux balance in complex networks. To a system with multiple cell types competing for nutrition, standard FBA is not directly applicable however, as cell numbers and thereby metabolic fluxes vary over time. \u003cem\u003eDynamic\u003c/em\u003e FBA (dFBA) allows fluxes to vary with time at time scales longer than required for metabolic relaxation inside the cells. Consequently, for each metabolic intermediate synthesis plus import should continue to equal degradation plus export fluxes\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e\u003c/span\u003e,\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e. dFBA typically uses kinetic equations for dominant nutrient supply rates as functions of concentrations of growth substrates outside the network. However, it does not address the impact of cell number variation with time\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e. The cell concentrations within the network should be variable as well as dependent on the dynamic nutrient concentrations.\u003c/p\u003e \u003cp\u003eDetailed kinetic modelling\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e\u003c/span\u003e,\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e should enable the modelling of metabolic networks with time variant cell numbers and nonlinearities of any type. However, kinetic modelling requires extensive kinetic details that are largely unknown for the topic at hand. Therefore, we here develop a method similar to FBA but capable of handling interacting and proliferating cell types with time-varying cell numbers. The new variant of FBA, which we shall refer to as dynamic cell-cell competition FBA (dcFBA) is applied to different cell types depending on each other through metabolites and/or growth factors. We find that such cross regulation needs to fulfill certain requirements in order to produce a stable organism consisting of the various cell types. Mutation to higher inherent growth rates should not in itself lead to tumorigenesis, but mutations in cross regulation should.\u003c/p\u003e"},{"header":"Results","content":"\u003cp\u003e1. Towards dynamic competitive FBA\u003c/p\u003e \u003cp\u003e \u003cb\u003e1\u0026ndash;1 Neither competition for common substrate nor metabolic dependence on common goods produces stable coexistence in standard FBA with total biomass as objective\u003c/b\u003e \u003c/p\u003e \u003cp\u003eIn this paper we shall examine whether two (or three, see below) cell types that compete for a common metabolic substrate can reach steady coexistence. Figure\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e shows the metabolic network that we used in the flux balance analysis. It allows for the two cell types also to depend on each other through \u0026lsquo;common goods\u0026rsquo; X (produced by cell type 1) and Y (produced by cell type 2) that both cell types require for their catabolism of glucose, their synthesis of X or Y and their growth (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003eA). Growth may have less than maximal yield because some glucose escapes to by-product; we call the corresponding flux rate ω. The scheme has 8 reactions, 5 metabolic intermediates (glucose, I1, I2, X, and Y) and one fixed flux (the glucose influx). A flux balance analysis (i.e. requiring steady state for the 5 intermediates) produced two modes of variation of the system. We selected the flux to by-product, and the difference between the biomass synthesis rates of cell type 2 and cell type 1 (which we called \u003cem\u003eβ\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003eA and \u003cem\u003eβ\u003c/em\u003e in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003eB) as variables to monitor those modes of variation (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003eA). With total biomass synthesis as objective function, the flux to by-product drops to zero (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003eB). Sections \u003cspan refid=\"Sec3\" class=\"InternalRef\"\u003e1\u003c/span\u003e\u0026ndash;\u003cspan refid=\"Sec6\" class=\"InternalRef\"\u003e3\u003c/span\u003e of the supplementary results show that the optimal flux always ran to the cheapest biomass, or to both cell types with different growth flux production. We conclude that competition for a common metabolic substrate and interdependence through common goods such as in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003eA, does not suffice to achieve coexistence of cell types.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFigure\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003eC and Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003eD remind us that the actual network structure should be more complicated as it should also require balances around the concentrations of biomass 1 and biomass 2 (see also section \u003cspan refid=\"Sec6\" class=\"InternalRef\"\u003e3\u003c/span\u003e\u0026thinsp;\u0026minus;\u0026thinsp;1 of Supplementary Results). These additions do not really affect to outcome of Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003eB however (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003eD): One still obtains the situation that the more growth rate of the cheapest cell type becomes persistently higher than the specific growth rate of the more expensive cell type, so that no coexistence arises.\u003c/p\u003e \u003cp\u003eFigure\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003eC lacks one aspect of reality however. If the biomass synthesis flux of cell type 1 is lower than that of cell type 2, whilst the sum of the two fluxes must equal the maximum 0.5, the required synthesis rates of X and Y remain 1/h. The amount of biomass of type 1 will however decrease with time, so that, as time goes on, the same amount of X has to be synthesized by less and less of cell type 1. The standard FBA that we used up to this point does not accommodate the expectation that there should be a maximum to the X synthesis per unit biomass of cell type 1. When that maximum is achieved the growth cell type 2 should stop, perhaps causing coexistence. In the next sections we develop a variant of FBA that addresses these issues.\u003c/p\u003e \u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003e1\u0026ndash;2 Preset exponential growth FBA for two cell types with capacity limitations\u003c/h2\u003e \u003cp\u003eWe first effected of the tendency of the two cell types to grow exponentially by presetting exponential growth equations, as detailed in the Methods. As shown in Supplementary Fig.\u0026nbsp;9, if cell type 2 had a higher inherent specific growth rate, it always outgrew the cell type 1: still no stable coexistence developed. Paradoxically, as cell type 1 disappeared and with it the capacity to synthesize the common good X, cell type 2 continued to grow even though a shortage of X should arise. The required capacity for X synthesis, i.e. the flux per unit cell 1, increased to infinity (Supplementary Figs.\u0026nbsp;9D-9F).\u003c/p\u003e \u003cp\u003eWe then instated a maximum capacity by giving the X synthesis reaction a flux bound of three times the concentration of cell type 1. Figure\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e shows the consequences of this limitation of metabolic capacities for the preset growth model. Initially, Biomass 2 again increased whilst Biomass 1 decreased and with it the upper bound for X synthesis. After t\u0026thinsp;=\u0026thinsp;12 h (for β(0)\u0026thinsp;=\u0026thinsp;0.05, Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003eA), the upper bound for X synthesis became lower than what was required, Biomass 2 abruptly decreased with time and the decrease with time of Biomass 1 accelerated. Total biomass thereby also decreased with time, ultimately to zero. For higher initial growth biases in favor of cell type 2, e.g. β(0)\u0026thinsp;=\u0026thinsp;0.1 (Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003eB) or β(0)\u0026thinsp;=\u0026thinsp;0.2 (Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003eC), the system behaved similarly, the transition happening earlier. The actual growth rate bias \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\left(t\\right)=\\frac{\\text{b}2\\left(\\text{t}\\right)-\\text{b}1\\left(\\text{t}\\right)}{2}\\)\u003c/span\u003e\u003c/span\u003e changed with time to zero as indicated in Supplementary Fig.\u0026nbsp;10: No stable coexistence was reached in any of the cases studied.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e1\u0026ndash;3 Stepwise growth FBA for two cell types\u003c/h2\u003e \u003cp\u003eWe also developed a \u0026lsquo;stepwise-growth FBA\u0026rsquo; algorithm, whereby at each time point the rate of biomass synthesis is related to the cell number. The points in Supplementary Figs, 11A-C display the predicted cell numbers as function of time for three growth rate biases β. Again, one cell type ultimately outcompeted the other cell type, at any growth bias.\u003c/p\u003e \u003cp\u003eWe then incorporated maximum production capacities of X and Y per unit biomass of cell types 1 and 2 (as specified under Methods). Figure\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e demonstrates the outcome for three values of β(0). Again (cf. Figure\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e) a greater growth bias difference led to a shorter lifespan for the system. Even when using the stepwise growth algorithm with maximum metabolic capacities of the two cell types, no stable coexistence was obtained.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eA kinetic model integrated by Copasi\u003csup\u003e\u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e32\u003c/span\u003e\u003c/sup\u003e did not reach coexistence either (Supplementary Fig.\u0026nbsp;12).\u003c/p\u003e \u003cp\u003e2. Regulation for preset-exponential growth and stepwise growth\u003c/p\u003e \u003cp\u003eHere, we will explore whether explicit regulatory mechanisms might furnish co-existence of cell types differing in specific growth rates.\u003c/p\u003e \u003cp\u003e \u003cb\u003e2\u0026thinsp;\u0026minus;\u0026thinsp;1 Preset-exponential growth FBA with capacity limits and cross regulation does not lead to stability either\u003c/b\u003e \u003c/p\u003e \u003cp\u003eNow we introduce direct cross-regulation between the two cell types: The biomass synthesis rate of cell type 1 (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({b}_{1,e,r ub}\\left(t\\right)\\)\u003c/span\u003e\u003c/span\u003e) will now be considered to be positively regulated (ε\u0026thinsp;\u0026gt;\u0026thinsp;0) by the number of cells of type 2, as represented by a factor \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\left({B}_{2,e,r}\\left(t\\right)\\right)}^{\\epsilon }\\)\u003c/span\u003e\u003c/span\u003e in the numerator (the negative dependence through the denominator continues to reflect the required Carbon flux balance, i.e. the competition for the glucose). The mechanism of this regulation might be a dependence of the expression level of an enzyme with much flux control on the specific growth rate of cell type 1. This dependence could involve gene expression regulation by a signal transduction route starting with a plasma membrane receptor addressed by a growth factor secreted by cell type 2 (see Methods). ε is then the \u0026lsquo;elasticity coefficient\u0026rsquo; of the regulation\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e33\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e. We explored the effect of different regulatory strengths (i.e. ε\u0026thinsp;=\u0026thinsp;0.5 or 1). Figure\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e shows that although this regulation postponed the time at which the system collapsed, it did not establish coexistence.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e2\u0026ndash;2 Stepwise growth FBA with cross regulation does show stability\u003c/h2\u003e \u003cp\u003eWe then added the same type of cross regulation to the stepwise FBA algorithm with capacity limitation, again with an adjustable regulation elasticity \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\epsilon\\)\u003c/span\u003e\u003c/span\u003e. Cell coexistence arose when the regulation power was strong enough (0.5 or 1; Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e). The purple and yellow lines recall that in the absence of such regulation (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\epsilon =0\\)\u003c/span\u003e\u003c/span\u003e), no such stable coexistence developed. The minimum regulation power (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\epsilon\\)\u003c/span\u003e\u003c/span\u003e value) for coexistence at three initial growth biases was 0.17 (β(0)\u0026thinsp;=\u0026thinsp;0.05), 0.36 (β(0)\u0026thinsp;=\u0026thinsp;0.1) and 0.92 (β(0)\u0026thinsp;=\u0026thinsp;0.2).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eIn the presence of the regulation, the steady state was reached before the maximum capacity was hit, so that the metabolic limitation was ineffective and even irrelevant for reaching coexistence. Accordingly, it was the cross regulation and not the limited metabolic capacities that made the two cell types co-exist. Complete regulation, i.e. an elasticity coefficient ε\u0026thinsp;=\u0026thinsp;1, sufficed to bring the two cell types into coexistence even if their inherent growth rates differed by a factor of 9. At a high inherent growth bias, stable co-existence required a strong regulation power (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003eC). Also, in the kinetic model co-existence could be achieved when such regulation was put in (see Supplementary Fig.\u0026nbsp;14).\u003c/p\u003e \u003cp\u003e3. Two social cell types and an a-social mutant\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003e3\u0026thinsp;\u0026minus;\u0026thinsp;1 An a-social mutant\u003c/h2\u003e \u003cp\u003eWe then introduced a third cell type again under consideration of growth optimization for total biomass synthesis, flux balance, regulation and limited metabolic capacity (Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e). We modeled the third cell type as a mutant of cell type 1 that continues to use common goods X and Y, that converts glucose to its intermediate I3 and that produces X, but has undergone a mutation interfering with the cross regulation with cell type 2. As a control we first considered a cell type 3 that was the same as type 1, i.e. neither autistic nor egoistic (Supplementary Fig.\u0026nbsp;16; \u0026lsquo;autistic\u0026rsquo; and \u0026lsquo;egoistic\u0026rsquo; mean, respectively, that the mutant is not responsive to regulation by cell type 2, or does not regulate cell type 2). The results in Supplementary Fig.\u0026nbsp;16 were the same as in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003eA with regulated power of 1. Then, there we considered three cases for the new mutant: 1) cell type 3 is both autistic and egoistic; 2) cell type 3 is autistic; 3) cell type 3 is egoistic.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eIn case \u003cspan refid=\"FPar1\" class=\"InternalRef\"\u003e1\u003c/span\u003e, the mutant cell type 3 outgrew the other two cell types, bringing both cell numbers of the latter two to zero and its own numbers subsequently. The outgrowth of the mutant cells occurred even though it had the same (or lower) inherent specific growth rate (0.2) as cell type1 and even lower than that of cell type2 (Supplementary Fig.\u0026nbsp;17 and Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e). In the \u0026lsquo;autism\u0026rsquo; case 2, the mutant cell type 3 co-existed with cell type 2, but outgrew cell type 1 (Supplementary Fig.\u0026nbsp;18). Since the mutant (cell type 3) has the same functionality as its original (cell type 1), it did not break the system. In the \u0026lsquo;egoism\u0026rsquo; case 3 the mutant could not outgrow the other two cell types and only existed at low concentrations, which may simulate a benign tumor (Supplementary Fig.\u0026nbsp;19). Next, we shall focus on case \u003cspan refid=\"FPar1\" class=\"InternalRef\"\u003e1\u003c/span\u003e for the three cell types system, with the stable situation of Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003eA with \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\epsilon }=1\\)\u003c/span\u003e\u003c/span\u003e as reference.\u003c/p\u003e \u003cp\u003e \u003cstrong\u003eCase 1\u003c/strong\u003e describes an \u0026lsquo;asocial cell type\u0026rsquo; in a system comprising of social cell types, should outgrow the other cell types. This might seem remarkable because the signals it stopped responding to, appear to be stimulatory signals: an increase in the number of type 2 cells was modelled as enhancing the growth of the cell. After the mutation this stimulation was lost, suggesting that the mutant should be worse and not better off. However, the result is on the contrary, which is due to the growth regulating factor, which should be less than 1 to control excessive cell growth and should be proportional to cell numbers.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec7\" class=\"Section2\"\u003e \u003ch2\u003e3\u0026thinsp;\u0026minus;\u0026thinsp;2 Three cell types regulating each other\u003c/h2\u003e \u003cp\u003eThe emergence of the transformed cell type that has completely lost all regulation through mutation is unlikely. Most breast cancers are still estrogen dependent, for instance\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e34\u003c/span\u003e\u003c/span\u003e,\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e35\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e. We therefore also considered the situation in which all three cell types continue to regulate each other at various elasticities. Specifically, we used the equations described in Methods to set the upper bound for biomass synthesis for each cell type and employed the stepwise growth dcFBA to determine the relationship between biomass synthesis, cell number and time. When we increased the reciprocal cross-regulation elasticity (\u0026#120574; value) between the normal cells of type 1 and type 2 on the one hand and the \u0026lsquo;transformed\u0026rsquo; cell type 3 on the other hand, we found that, already at the moderate regulation of \u0026#120574;=0.5, cell type 3 no longer outgrew the other two cell types (Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e). For an initial population composition of B\u003csub\u003e1\u003c/sub\u003e(0)\u0026thinsp;=\u0026thinsp;0.39, B\u003csub\u003e2\u003c/sub\u003e(0)\u0026thinsp;=\u0026thinsp;0.59 and B\u003csub\u003e3\u003c/sub\u003e(0)\u0026thinsp;=\u0026thinsp;0.02, the minimal magnitude of \u0026#120574; for stability was around 0.3, at a death rate of 0.5/h. The cause of out-competing is linked to a lack of regulatory mechanisms in winner cells. Additionally, we investigated the higher growth rate of cell type 3 with and without regulation from other two cell types. As illustrated in Fig. S20, we found that cell type 3 is able to thrive at a high growth rate when regulation from the other two cell types is active, leading to the observed coexistence. Again, this result shows that cell regulation, rather than its inherent growth rate, is the determining factor for cells out-competing each other.\u003c/p\u003e \u003cp\u003e \u003cb\u003e3\u0026ndash;3 Only cell type 3 (mutant) being regulated by the other two cell types (normal cells)\u003c/b\u003e \u003c/p\u003e \u003cp\u003eIn actual situations, the transformed cell is still reliant on the normal cells\u0026rsquo; activities, such as a tumor cell depending on the lung and the heart cell. However, the tumor cell may not support the normal cells in any way. When increasing the cross-regulation (\u0026#120574; value) between the normal cells and the transformed cell, the transformed cell initially outgrew the other two cell types, and then this could subsequently lead to a coexistence of the three cell types (Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003eC). However, this coexistence is not a typical occurrence since more often the normal cell count drastically decreases and approaches zero. As the regulation strength was increased to 0.4 or more, the two normal cell types outgrew the transformed cell (as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003eD): if we increased the regulation power \u0026#120574; above 0.5.\u003c/p\u003e \u003c/div\u003e"},{"header":"Discussion","content":"\u003cp\u003eA new, \u0026lsquo;dynamic competitive FBA\u0026rsquo; (dcFBA) methodology was developed to investigate the behavior of a multicellular system over time. Mathematical equations set the reaction bounds for a nutritionally competitive situation. Small time intervals were employed to iterate dynamic progress and flux balance requirement. For two cell types with different inherent growth rates, regulation was necessary to prevent one cell type from going extinct. In a three-cell type system involving two normal cell types and a transformed cell type, the loss of regulation led to the emergence of \u0026lsquo;asocial\u0026rsquo; cells that exploited resources from others. We showed that this a-sociality and not a higher intrinsic growth rate was responsible for the mutant outgrowing the normal cells and endangering the persistence of the multicellular organism. The dcFBA developed here may also be useful for other issues around tissue homeostasis, such as the control of liver size. Within minutes after partial surgical hepatectomy, mammalian liver demonstrated regenerative abilities, recovering to its original organ mass within a matter of weeks\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e36\u003c/span\u003e,\u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e37\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e. The liver is also capable of returning to its normal size after hypertrophy and/or hyperplasia, by the activation of regression mechanisms (e.g. cell apoptosis), using mechanisms monitoring cell number or cell size\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e38\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e. Another example may reside in the competition between astrocytes and neurons, e.g. for amino acids uptake during brain development. In a previous paper\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR39\" class=\"CitationRef\"\u003e39\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e, we developed another mode of competitive FBA to demonstrate implications of the amino acid competition for uptake across the blood-brain barrier. One may now fruitfully combine the two new types of FBA and include neurotransmitter cross-talk and metabolism.\u003c/p\u003e \u003cp\u003eFBA differs from kinetic modelling in requiring virtually no kinetic detail\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e. Since it is these kinetic details that are still missing for the complex networks at hand, this should make the dcFBA approach a useful addition to the modelling toolbox. FBA\u0026rsquo;s prediction of much fewer than all possible steady state behaviors, requires the assumption that network behavior is optimal however. The optimality is usually assumed to be maximal growth rate of the cells in question\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e and that is what we also assumed here: maximality of \u003cem\u003etotal\u003c/em\u003e biomass synthesis.\u003c/p\u003e \u003cp\u003eSection 10 of the supplementary material shows that choosing the synthesis rate of either cell type 1 or cell type 2 as objective function, does not affect the essence of our conclusions.\u003c/p\u003e \u003cp\u003eOur study did not deliver detail. The integration of more molecular, signaling and interaction information into FBA-kinetics hybrid models may lead to a more realistic representation of metabolic and signaling phenomena. MYC-mediated cell competition for instance is prevalent across different organs and tissues, including fibroblasts\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR40\" class=\"CitationRef\"\u003e40\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e and heart cells\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR41\" class=\"CitationRef\"\u003e41\u003c/span\u003e\u003c/span\u003e,\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR42\" class=\"CitationRef\"\u003e42\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e, and plays a role in the growth and expansion of cancer cells \u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR43\" class=\"CitationRef\"\u003e43\u003c/span\u003e\u003c/span\u003e,\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR44\" class=\"CitationRef\"\u003e44\u003c/span\u003e\u003c/span\u003e,\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR45\" class=\"CitationRef\"\u003e45\u003c/span\u003e\u003c/span\u003e,\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR46\" class=\"CitationRef\"\u003e46\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e. Overexpression of MYC transforms tumor cells into super-competitors, enabling them to eliminate adjacent wild-type cells\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR47\" class=\"CitationRef\"\u003e47\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e. The growth signaling pathway in out-competed cells becomes impaired\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR48\" class=\"CitationRef\"\u003e48\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e, as evidenced by the decreased decapentaplegic transduction observed in out-competed cells\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR49\" class=\"CitationRef\"\u003e49\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e: the out-competed cells capture fewer growth factors. The winner cells increase their engulfment\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e, acquiring more space and resources while evading regulatory mechanisms from normal cells. Tumor cells also inactivate the Hippo pathway, thereby promoting their own proliferation\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR50\" class=\"CitationRef\"\u003e50\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e. The dcFBA developed here may be extended to deal with some of this complexity.\u003c/p\u003e \u003cp\u003eCompetition for nutrients, space, and growth factors is ubiquitous in multicellular organisms. On the one hand, normal or young cells compete with damaged or older cells to maintain tissue homeostasis; preventing such competition may result in lymphoblastic leukemia\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR51\" class=\"CitationRef\"\u003e51\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e. On the other hand, tumor or super-competitor cells become asocial when engaging in cell competition, leading to the elimination of normal cells. Our findings suggest that the critical time (i.e., the time before the tumor grows to a critical size) can be prolonged by increasing the initial cell number of normal cells (cell type 1 and type 2) (Supplementary Fig.\u0026nbsp;22). This may imply that an active body that has a low reduced glucose (metabolic substrate) level might delay the growth of tumor cells, suggesting an alternative to glucose fasting\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR52\" class=\"CitationRef\"\u003e52\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e. That also the cell-death rate constant (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{D}\\)\u003c/span\u003e\u003c/span\u003e) affected the critical time (Supplementary Fig.\u0026nbsp;23), suggests that elderly individuals may be more sensitive to competition by tumor cells. Our results may further inspire the development of drugs or therapies targeting the competition or cross-regulation rather than cytotoxic agents or growth rate inhibitors: a higher inherent specific growth rate is unlikely to be the primary factor driving tumor cell growth (Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e and Supplementary Fig.\u0026nbsp;20). When social cells with a higher inherent specific growth rate were present, the asocial cell type (cell type 3) was able to outgrow them, but not when the latter was social and had a \u003cem\u003ehigher\u003c/em\u003e specific growth rate than other cell types.\u003c/p\u003e \u003cp\u003eOur observations of the effects of cell-cell competition and cross-regulation may also be relevant more generally: Also, the components of an ecosystem are interdependent and single \u0026lsquo;dominant\u0026rsquo; species or mutants rarely thrive in isolation. Relevant scenarios in global warming or human-induced pollution may be analyzed by our new dcFBA methodology with a proper objective function.\u003c/p\u003e "},{"header":"Methods","content":"\u003cp\u003e1 Model building\u0026mdash;Two cell types competing for common free-energy/carbon substrate and interacting through common goods.\u003c/p\u003e \u003cp\u003eIn all our models two (or three) cell types use glucose to produce their own metabolic intermediate (I1 and I2, respectively) (Supplementary Fig.\u0026nbsp;4). The so-called \u0026lsquo;common goods\u0026rsquo; X and Y are produced by cell type 1 and 2, respectively, from their metabolic intermediate, and are both required by both cell types to produce their metabolic intermediate. Each cell type uses its intermediate either to produce its own biomass (i.e. to increase its cell number) or to produce one common good. The model also contains a by-product reaction from glucose and a carbon dioxide secreting reaction. In terms of Carbon, our models transduce half the glucose into biomass and the other half into carbon dioxide, a carbon/carbon yield that is not unusual\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR53\" class=\"CitationRef\"\u003e53\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e. The stoichiometries for the GTI1 reaction (see Supplementary Fig.\u0026nbsp;4) are taken to be -1 for glucose, X, and Y and +\u0026thinsp;1 for I1. For the GTI2 reaction they are analogously \u0026minus;\u0026thinsp;1 for glucose, X and Y and +\u0026thinsp;1 for I2. For DI1X they are \u0026minus;\u0026thinsp;1, +\u0026thinsp;4, and +\u0026thinsp;1 for I1, X, and CO\u003csub\u003e2\u003c/sub\u003e, respectively. For DI2Y they are \u0026minus;\u0026thinsp;1, +\u0026thinsp;4, and +\u0026thinsp;1 for I2, Y, and CO\u003csub\u003e2\u003c/sub\u003e, respectively. For the reaction towards biomass 1 they are \u0026minus;\u0026thinsp;1 and +\u0026thinsp;1 (or 1/b) for the intermediate I1 and the biomass 1, respectively. For synthesis of biomass 2, these numbers are \u0026minus;\u0026thinsp;1 and +\u0026thinsp;1 for the intermediate I2 and the biomass 2, respectively. In our simple model, only glucose was supplied in the medium. This model may correspond to the interaction between \u0026lsquo;lung cells\u0026rsquo; and \u0026lsquo;liver cells\u0026rsquo;, which both depend for their growth on externally supplied glucose (for Carbon), glutamine (for nitrogen) and oxygen (for energetics) in the circulating blood. Lung cells oxygenate the blood, whilst liver cells enrich it with glutamine. Both cell types may use the glucose, glutamine and oxygen in growth and respiration producing carbon dioxide and urea.\u003c/p\u003e \u003cp\u003eBy using the COBRA routine\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR54\" class=\"CitationRef\"\u003e54\u003c/span\u003e\u003c/span\u003e,\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR55\" class=\"CitationRef\"\u003e55\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e, FBA was performed with the sum of the two biomass production fluxes as objective, and the maximum total biomass synthesis flux was determined. This produced flux balances for glucose, X and Y, as well as I1 and I2. Acknowledging that COBRA\u0026rsquo;s FBA only yields a single flux pattern also when there are multiple equivalent patterns, we employed Flux Variability Analysis (FVA)\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan citationid=\"CR56\" class=\"CitationRef\"\u003e56\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e to compute the range of possible flux patterns for the same maximum magnitude of the objective function, by limiting each step to smaller absolute values. We also checked the effect of making biomass 1 \u0026lsquo;more expensive\u0026rsquo; by making the production of biomass1 cost 2 units of I1, and biomass 2 \u0026lsquo;cheaper\u0026rsquo; by requiring only one unit of I2 to produce 1 unit of biomass 2. In other models the biomass 1 synthesis reaction only produce 1/m C-moles of biomass per C-mole of I1.\u003c/p\u003e \u003cp\u003e2 Growth FBA\u0026mdash;Dynamic cell competition Flux Balance Analysis for two cell types competing for a common substrate and with time varying cell densities, without or with capacity limitations, and with cross dependence through common goods X and Y\u003c/p\u003e \u003cp\u003eAccording to the FBA, the biomass synthesis fluxes should optimally add up to 0.5 (This was because we assumed the glucose uptake equaled 1. In case where the glucose uptake differs from this value, the same equations should be applied with appropriate adjustment.). Assuming that this optimum is achieved at all times, i.e. also as the biomass concentration ratios change, the biomass synthesis fluxes for biomass 1 and biomass 2, respectively, should tend towards:\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$${b}_{1,e,ub}\\left(t\\right)=0.5\\bullet \\frac{{\\mu }_{1}\\bullet {f}_{1}\\left(t\\right)}{{\\mu }_{1}\\bullet {f}_{1}\\left(t\\right)+{\\mu }_{2}{\\bullet f}_{2}\\left(t\\right)}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$${b}_{2,e,ub}\\left(t\\right)=0.5\\bullet \\frac{{\\mu }_{2}\\bullet {f}_{2}\\left(t\\right)}{{\\mu }_{1}\\bullet {f}_{1}\\left(t\\right)+{\\mu }_{2}{\\bullet f}_{2}\\left(t\\right)}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewith arbitrary functions \u003cem\u003ef\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e(t)\u003c/em\u003e and \u003cem\u003ef\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e(t)\u003c/em\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\mu }_{1} \\text{a}\\text{n}\\text{d} {\\mu }_{2}\\)\u003c/span\u003e\u003c/span\u003e representing the inherent specific growth rates.\u003c/p\u003e \u003cp\u003e \u003cb\u003e2\u0026thinsp;\u0026minus;\u0026thinsp;1 Preset-exponential growth FBA for two cell types with cross dependence through common goods, obeying the optimal FBA condition of constant optimal total biomass flux\u003c/b\u003e \u003c/p\u003e \u003cp\u003eAs growth of cells tends to be exponential, we assume exponential functions of time for the two growth tendencies \u003cem\u003ef(t)\u003c/em\u003e, i.e.:\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$${b}_{1,e,ub}\\left(t\\right)= 0.5\\bullet \\frac{{\\mu }_{1}\\bullet {e}^{\\frac{{\\mu }_{1}\\bullet t}{{\\mu }_{1}+{\\mu }_{2}}}}{{\\mu }_{1}\\bullet {e}^{\\frac{{\\mu }_{1}\\bullet t}{{\\mu }_{1}+{\\mu }_{2}}}+{\\mu }_{2}\\bullet {e}^{\\frac{{\\mu }_{2}\\bullet t}{{\\mu }_{1}+{\\mu }_{2}}}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$${b}_{2,e,ub}\\left(t\\right)= 0.5\\bullet \\frac{{\\mu }_{2}\\bullet {e}^{\\frac{{\\mu }_{2}\\bullet t}{{\\mu }_{1}+{\\mu }_{2}}}}{{\\mu }_{1}\\bullet {e}^{\\frac{{\\mu }_{1}\\bullet t}{{\\mu }_{1}+{\\mu }_{2}}}+{\\mu }_{2}\\bullet {e}^{\\frac{{\\mu }_{2}\\bullet t}{{\\mu }_{1}+{\\mu }_{2}}}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003e \u003cem\u003e\u0026micro;\u003c/em\u003e \u003csub\u003e \u003cem\u003e1\u003c/em\u003e \u003c/sub\u003e\u0026thinsp;=\u0026thinsp;0.25-\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\beta\\)\u003c/span\u003e\u003c/span\u003e and \u003cem\u003e\u0026micro;\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;0.25+\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\beta\\)\u003c/span\u003e\u003c/span\u003e are the specific growth rates for cell types 1 and 2 and assumed to be constant. We used the \u0026lsquo;\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\frac{{\\mu }_{1}}{{\\mu }_{1}+{\\mu }_{2}}\\)\u003c/span\u003e\u003c/span\u003e\u0026rsquo; and \u0026lsquo;\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\frac{{\\mu }_{2}}{{\\mu }_{1}+{\\mu }_{2}}\\)\u003c/span\u003e\u003c/span\u003e\u0026rsquo; to normalize the growth rates for both cell types here. The parameter \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\beta\\)\u003c/span\u003e\u003c/span\u003e is the initial growth rate bias in favor of cell type 2. Because the growth rate is set \u003cem\u003ea priori\u003c/em\u003e as an exponential function of time we call this the \u0026lsquo;preset-exponential FBA\u0026rsquo; procedure.\u003c/p\u003e \u003cp\u003eA familiar way to make FBA produce maximal flux through a reaction is to give the corresponding reaction an upper bound equal to that flux and all other reactions higher upper bounds. We therefore set the upper bounds for the biomass synthesis reactions equal to the above expressions and then carried out FBA for Supplementary Fig.\u0026nbsp;4 at subsequent time points, separated from each other by \u003cem\u003ets\u003c/em\u003e (a small (infinitesimal) amount of time units) and with glucose efflux 1/\u003cem\u003ets\u003c/em\u003e. This produced \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\text{b}\\text{1}\\text{,e}\\text{,FBA}\\left(t\\right)\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\text{b}\\text{2,}\\text{e,FBA}\\left(t\\right)\\)\u003c/span\u003e\u003c/span\u003e, which were indeed identical to the upper bounds. Acknowledging that the cells should also be subject to death processes (for which we used a first order process with rate constant \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003eD\u003c/em\u003e\u003c/sub\u003e (we chose \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003eD\u003c/em\u003e\u003c/sub\u003e to equal 0.5)), we calculated the Biomass concentrations for the two cell types at each time point from:\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$${B}_{1,e}\\left(t\\right)={B}_{1,e}\\left(t-ts\\right)\\bullet \\left(1-ts\\bullet {k}_{D}\\right)+ts\\bullet \\text{b}\\text{1}\\text{,e,FBA}\\left(t-ts\\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e\n$${B}_{2,e}\\left(t\\right)={B}_{2,e}\\left(t-ts\\right)\\bullet \\left(1-ts\\bullet {k}_{D}\\right)+ts\\bullet \\text{b}\\text{2,}\\text{e,FBA}\\left(t-ts\\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cem\u003ets\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.1; B\u003csub\u003e1\u003c/sub\u003e(0)\u0026thinsp;=\u0026thinsp;0.4 and B\u003csub\u003e2\u003c/sub\u003e(0)\u0026thinsp;=\u0026thinsp;0.6 for \u003cem\u003eβ\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.05; B\u003csub\u003e1\u003c/sub\u003e(0)\u0026thinsp;=\u0026thinsp;0.3 and B\u003csub\u003e2\u003c/sub\u003e(0)\u0026thinsp;=\u0026thinsp;0.7 for \u003cem\u003eβ\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.1; B\u003csub\u003e1\u003c/sub\u003e(0)\u0026thinsp;=\u0026thinsp;0.1 and B\u003csub\u003e2\u003c/sub\u003e(0)\u0026thinsp;=\u0026thinsp;0.9 for \u003cem\u003eβ\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.2 (These values will be the same for subsequent calculations unless specifically stated). We chose these values because we want the whole system to be stable before system broken (new biomass flux equaling the death flux). The difference between the cell numbers at the beginning is in accordance with their relative growth rates.\u003c/p\u003e\n\u003ch3\u003e2–2 Stepwise growth FBA for two cell types\u003c/h3\u003e\n\u003cp\u003eWe next developed an FBA growth algorithm in which the growth kinetics would not be pre-defined. At the glucose branch we consider fluxes towards I1, I2 and an overflow flux \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\omega }\\)\u003c/span\u003e\u003c/span\u003e, the rates of which we assume to be all proportional to the glucose concentration level. The biomass synthesis ratio of cell types will be proportional to their cell number (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\frac{{b}_{1,s,ub}\\left(t\\right)}{{b}_{2,s,ub}\\left(t\\right)}=\\frac{{\\mu }_{1}\\bullet {B}_{1,s}\\left(t-ts\\right)}{{\\mu }_{2}\\bullet {B}_{2,s}\\left(t-ts\\right)}\\)\u003c/span\u003e\u003c/span\u003e) described as differing in \u003cem\u003e\u0026micro;\u003c/em\u003e, but this could just as well reflect a difference in yield. With the common good balance met and the glucose influx fixed to 1, the two fluxes towards biomass amount to:\u003cdiv id=\"Equ7\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ7\" name=\"EquationSource\"\u003e\n$${b}_{1,s,ub}\\left(t\\right)=\\frac{{0.5\\bullet \\mu }_{1}\\bullet {B}_{1,s}\\left(t-ts\\right)\\bullet \\left(1-{\\omega }\\right)}{{\\mu }_{1}\\bullet {B}_{1,s}\\left(t-ts\\right)+{\\mu }_{2}\\bullet {B}_{2,s}\\left(t-ts\\right)}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e7\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ8\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ8\" name=\"EquationSource\"\u003e\n$${b}_{2,s,ub}\\left(t\\right)=\\frac{{0.5\\bullet \\mu }_{2}\\bullet {B}_{2,s}\\left(t-ts\\right)\\bullet \\left(1-{\\omega }\\right)}{{\\mu }_{1}\\bullet {B}_{1,s}\\left(t-ts\\right)+{\\mu }_{2}\\bullet {B}_{2,s}\\left(t-ts\\right)}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e8\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eBecause the growth rates are reset at every time point of computation, we call this the \u0026lsquo;stepwise-growth FBA\u0026rsquo; procedure. The amount of X required is equal to half of the sum of these two fluxes. Hence the fluxes from glucose to I1 and I2:\u003cdiv id=\"Equ9\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ9\" name=\"EquationSource\"\u003e\n$${v}_{1,g}\\left(t\\right)=\\frac{1-{\\omega }}{4}+\\frac{{0.5\\bullet \\mu }_{1}\\bullet {B}_{1,s}\\left(t-ts\\right)\\bullet \\left(1-{\\omega }\\right)}{{\\mu }_{1}\\bullet {B}_{1,s}\\left(t-ts\\right)+{\\mu }_{2}\\bullet {B}_{2,s}\\left(t-ts\\right)}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e9\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ10\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ10\" name=\"EquationSource\"\u003e\n$${v}_{2,g}\\left(t\\right)=\\frac{1-{\\omega }}{4}+\\frac{{0.5\\bullet \\mu }_{2}\\bullet {B}_{2,s}\\left(t-ts\\right)\\bullet \\left(1-{\\omega }\\right)}{{\\mu }_{1}\\bullet {B}_{1,s}\\left(t-ts\\right)+{\\mu }_{2}\\bullet {B}_{2,s}\\left(t-ts\\right)}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e10\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eTotal biomass synthesis equals:\u003cdiv id=\"Equ11\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ11\" name=\"EquationSource\"\u003e\n$${b}_{1,s,ub}\\left(t\\right)+{b}_{2,s,ub}\\left(t\\right)=\\frac{1-{\\omega }}{2}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e11\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhen asking for maximal total biomass synthesis, and if the metabolic capacities are unlimited, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\omega }\\)\u003c/span\u003e\u003c/span\u003e becomes equal to zero and disappears from the equations. At every time point FBA was carried out for the metabolic network of Supplementary Fig.\u0026nbsp;4, with \u003cem\u003eb\u003c/em\u003e\u003csub\u003e\u003cem\u003e1,s,ub\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003eb\u003c/em\u003e\u003csub\u003e\u003cem\u003e2,s,ub\u003c/em\u003e\u003c/sub\u003e as the upper bounds for the biomass synthesis for cell type 1 and 2, respectively: this produced the biomass synthesis fluxes \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\text{b}\\text{1,}\\text{s,FBA}\\left(t\\right)\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\text{b}\\text{2,}\\text{s,FBA}\\left(t\\right)\\)\u003c/span\u003e\u003c/span\u003e, which were effectively equal to the upper bounds. At each time step the cell concentrations were calculated as the same as above from with biomass synthesis fluxes and the death rates.\u003c/p\u003e\n\u003ch3\u003e2–3 Limiting metabolic capacities\u003c/h3\u003e\n\u003cp\u003eThe FBA calculations using the methods specified up to this point did not acknowledge any possible capacity limitation for producing common good X or Y that would arise due to lack of sufficient cells of type 1 or 2, respectively: the model continued to predict biomass synthesis of cell type 2 even when cell type 1, the sole producer of the X required by cell type 2 for the synthesis of its intermediate I2 and thereby for its growth, had been outgrown. To address this issue, we adjusted the upper bound for X and Y production so as to be proportional to the cell numbers of type 1 and type 2, respectively. Specifically, we set the maximum abilities to produce X or Y to three times their respective calculated cell numbers. After setting the reaction bound, we again ran the FBA at every time point to compute the growth rates of two cell types (i.e. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\text{b}\\text{1,}\\text{s}\\text{,FBA}\\left(t\\right)\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\text{b}\\text{2,}\\text{s,FBA}\\left(t\\right)\\)\u003c/span\u003e\u003c/span\u003e). By the usual multiplication of the biomass production fluxes by the duration of the time step and by correction for cell death, we then calculated the predicted cell numbers for each cell type. In an alternative methodology, 1-\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\omega }\\)\u003c/span\u003e\u003c/span\u003e in the above equations was replaced by:\u003c/p\u003e\n\u003cdiv id=\"Equ12\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ12\" name=\"EquationSource\"\u003e$$1-{\\omega }\\left(t\\right)=minimum(1;{ 4\\bullet {V}_{max}\\bullet B}_{1}\\left(t-ts\\right); {4\\bullet {V}_{max}\\bullet B}_{2}\\left(t-ts\\right))$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e12\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eWe also developed a kinetic model based on irreversible mass action rate equations and used Copasi\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan class=\"CitationRef\"\u003e32\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e to simulate the cells\u0026rsquo; growth dynamics (See supplementary material).\u003c/p\u003e\n\u003cp\u003e3 Regulation\u003c/p\u003e\n\u003cdiv id=\"Sec11\" class=\"Section2\"\u003e\n \u003ch2\u003e3\u0026thinsp;\u0026minus;\u0026thinsp;1 Regulation in preset-exponential growth FBA\u003c/h2\u003e\n \u003cp\u003eWe also considered cases where the two cell types depend on each other for their growth also more directly than through the common goods X and Y. We assumed that cell type 2 produced a growth factor that stimulated growth of cell type 1 without being consumed by it, and that the concentration of that growth factor was proportional to the number of cells of type 2. The regulation is assumed to depend on the activation of a receptor by the binding of growth factor:\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec12\" class=\"Section2\"\u003e\n \u003ch2\u003eG\u0026thinsp;+\u0026thinsp;R⇋RG (13)\u003c/h2\u003e\n \u003cp\u003eWhere G is the growth factor produced by cell type 2 and R is the corresponding receptor in the plasma membrane of cell type 1. At binding equilibrium:\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec13\" class=\"Section2\"\u003e\n \u003cp\u003eRF=\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\frac{\\left[\\text{R}\\text{G}\\right]}{\\left[\\text{R}\\right]+\\left[\\text{R}\\text{G}\\right]}=\\frac{\\left[\\text{G}\\right]}{\\left[\\text{G}\\right]+{K}_{d}}\\)\u003c/span\u003e\u003c/span\u003e (14)\u003c/p\u003e\n \u003cp\u003ewhere the regulation factor (RF) is the fraction of receptor bound to growth factor G; \u003cem\u003eK\u003c/em\u003e\u003csub\u003e\u003cem\u003ed\u003c/em\u003e\u003c/sub\u003e is the dissociation equilibrium constant. If \u0026alpha; is the ratio of the production rate constant to the first-order dilution rate constant of G, then:\u003c/p\u003e\n \u003cdiv id=\"Equ13\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ13\" name=\"EquationSource\"\u003e$$\\text{G}={\\alpha }\\cdot {B}_{2}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e15\u003c/div\u003e\n \u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec14\" class=\"Section2\"\u003e\n \u003cp\u003eRF=\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\frac{{\\alpha }\\cdot {B}_{2}}{{\\alpha }\\cdot {B}_{2}+{K}_{d}}\\)\u003c/span\u003e\u003c/span\u003e=\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\frac{{B}_{2}}{{B}_{2}+\\frac{{K}_{d}}{{\\alpha }}}\\)\u003c/span\u003e\u003c/span\u003e (16)\u003c/p\u003e\n \u003cp\u003eThe regulation factor will always be smaller than 1 and depend on B\u003csub\u003e2\u003c/sub\u003e. We took a case where B\u003csub\u003e2\u003c/sub\u003e\u0026lt;\u0026lt;\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\frac{{K}_{d}}{{\\alpha }}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({K}_{d}={\\alpha }\\)\u003c/span\u003e\u003c/span\u003e, so that RF equaled B\u003csub\u003e2\u003c/sub\u003e, i.e. for simplicity we used B\u003csub\u003e2\u003c/sub\u003e (cell 2 number) to represent the activity factor with different strengths. We ensured that B\u003csub\u003e2\u003c/sub\u003e ranged between 0 and 1, by adjusting the unit for cell numbers (i.e. to a billion).\u003c/p\u003e\n \u003cp\u003eTo simulate the regulation, we made the rate of synthesis of cell type 1 proportional to RF (and hence to the concentration of cell type 2) taken to the power of the \u0026lsquo;elasticity\u0026rsquo; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\epsilon\\)\u003c/span\u003e\u003c/span\u003e:\u003c/p\u003e\n \u003cdiv id=\"Equ14\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ14\" name=\"EquationSource\"\u003e$${b}_{1,e,r ub}\\left(t\\right)= 0.5\\bullet \\frac{(0.25-)\\bullet {e}^{\\frac{\\left(0.25-\\right)\\bullet t}{0.5}} \\bullet {\\left({B}_{2,e,r}\\left(t\\right)\\right)}^{\\epsilon }}{(0.25-){\\bullet e}^{\\frac{\\left(0.25-\\right)\\bullet t}{0.5}}\\bullet {\\left({B}_{2,e,r}\\left(t\\right)\\right)}^{\\epsilon }+(0.25+)\\bullet {e}^{\\frac{\\left(0.25+\\right)\\bullet t}{0.5}}\\bullet {\\left({B}_{1,e,r}\\left(t\\right)\\right)}^{\\epsilon }}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e17\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u0026nbsp;\u003cspan class=\"mathinline\"\u003e\\(\\epsilon\\)\u003c/span\u003e\u0026nbsp;\u003c/span\u003e is the elasticity coefficient of the regulation \u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan class=\"CitationRef\"\u003e33\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e akin to the power in Biochemical Systems Theory\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan class=\"CitationRef\"\u003e57\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e. The corresponding dependence of synthesis of cell type 2 on the concentration of cell type 1 reads as follows:\u003c/p\u003e\n \u003cdiv id=\"Equ15\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ15\" name=\"EquationSource\"\u003e$${b}_{2,e,r,ub}\\left(t\\right)= 0.5\\bullet \\frac{(0.25+)\\bullet {e}^{\\frac{\\left(0.25+\\right)\\bullet t}{0.5}}\\bullet {\\left({B}_{1,e,r}\\left(t\\right)\\right)}^{\\epsilon }}{(0.25-){\\bullet e}^{\\frac{\\left(0.25-\\right)\\bullet t}{0.5}}\\bullet {\\left({B}_{2,e,r}\\left(t\\right)\\right)}^{\\epsilon }+(0.25+)\\bullet {e}^{\\frac{\\left(0.25+\\right)\\bullet t}{0.5}}\\bullet {\\left({B}_{1,e,r}\\left(t\\right)\\right)}^{\\epsilon }}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e18\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003eAs before we used these equations to set the upper bounds for the biomass synthesis reactions in the metabolic scheme and then carried out FBA with the total biomass synthesis as objective. The resulting biomass synthesis for both cell types were again used to calculate the cell levels for the next time point as same as above. As the equation contains a predefined exponential growth tendency, we call this procedure \u0026lsquo;regulated preset-exponential growth FBA\u0026rsquo;.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec15\" class=\"Section2\"\u003e\n \u003ch2\u003e3\u0026thinsp;\u0026minus;\u0026thinsp;2 Regulation in stepwise growth FBA\u003c/h2\u003e\n \u003cp\u003eWe also introduced cross regulation between cell types, at a strength again indicated by the parameter \u0026lsquo;\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\epsilon\\)\u003c/span\u003e\u003c/span\u003e\u0026rsquo; in the \u0026lsquo;stepwise growth\u0026rsquo; FBA model:\u003c/p\u003e\n \u003cdiv id=\"Equ16\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ16\" name=\"EquationSource\"\u003e$${b}_{1,s,r,ub}\\left(t\\right)=\\frac{0.5{\\bullet \\mu }_{1}\\bullet \\left({B}_{1,s,r}\\left(t-ts\\right)\\right)\\bullet {\\left({B}_{2,s,r}\\left(t-ts\\right)\\right)}^{\\epsilon }}{{\\mu }_{1}\\bullet \\left({B}_{1,s,r}\\left(t-ts\\right)\\right)\\bullet {\\left({B}_{2,s,r}\\left(t-ts\\right)\\right)}^{\\epsilon }+{\\mu }_{2}\\bullet {\\left({B}_{1,s,r}\\left(t-ts\\right)\\right)}^{\\epsilon }\\bullet \\left({B}_{2,s,r}\\left(t-ts\\right)\\right)}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e19\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ17\" class=\"Equation\"\u003e\u003cdiv class=\"mathdisplay\" id=\"FileID_Equ17\" name=\"EquationSource\"\u003e$${b}_{2,s,r,ub}\\left(t\\right)=\\frac{0.5{\\bullet \\mu }_{2}\\bullet {\\left({B}_{1,s,r}\\left(t-ts\\right)\\right)}^{\\epsilon }\\bullet \\left({B}_{2,s,r}\\left(t-ts\\right)\\right)}{{\\mu }_{1}\\bullet \\left({B}_{1,s,r}\\left(t-ts\\right)\\right)\\bullet {\\left({B}_{2,s,r}\\left(t-ts\\right)\\right)}^{\\epsilon }+{\\mu }_{2}\\bullet {\\left({B}_{1,s,r}\\left(t-ts\\right)\\right)}^{\\epsilon }\\bullet \\left({B}_{2,s,r}\\left(t-ts\\right)\\right)}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e20\u003c/div\u003e\u003c/div\u003e\u003cp\u003eHere \u003cem\u003eb\u003c/em\u003e\u003csub\u003e\u003cem\u003e1,s,r,ub\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003eb\u003c/em\u003e\u003csub\u003e\u003cem\u003e2,s,r,ub\u003c/em\u003e\u003c/sub\u003e are again used to set the upper bounds for the biomass synthesis reactions. After that, we get actual biomass synthesis rates for cell type 1 and 2 respectively with regulation in stepwise growth. And we use these values to calculate the cell number for cell type 1 and 2 like above. Although the inherent specific growth rates for cell types 1 and 2 are \u003cem\u003e\u0026micro;\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;0.25-\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\beta\\)\u003c/span\u003e\u003c/span\u003e and \u003cem\u003e\u0026micro;\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;0.25+\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\beta\\)\u003c/span\u003e\u003c/span\u003e in the beginning, respectively, the two actual specific growth rate tendencies may depend on time. We tried various regulation powers (i.e., different values of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\epsilon\\)\u003c/span\u003e\u003c/span\u003e) to set the reaction upper bound for biomass production and to check how the cell number was changing with time. As the equation does not contain a predefined \u003cem\u003eexponential\u003c/em\u003e growth tendency for the two cells types but this growth tendency is set by the stepwise change in biomass concentrations with time, we call this procedure \u0026lsquo;regulated stepwise growth FBA\u0026rsquo;.\u003c/p\u003e\u003cp\u003e4. Three cell types\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec16\" class=\"Section2\"\u003e\u003ch2\u003e4\u0026thinsp;\u0026minus;\u0026thinsp;1 Stepwise growth FBA for two social cell types together with one asocial cell type\u003c/h2\u003e\u003cp\u003eAfter calculating the system with two cell types, we added another cell type to the system. The third cell type was a mutant of cell type 1 deficient in the communication with cell type 2. Otherwise it functioned identically to cell type 1: It still had the ability to produce common good X, and used common goods X and Y whilst converting glucose to its intermediate metabolite I3. The model building file is provided in the folder \u0026lsquo;files\u0026rsquo; in the GitHub directory. The influx was again 1 C-mole/h, the sum of biomass 1, biomass 2 and biomass 3 synthesis was again 0.5 C-mole/h. The excess carbon produced in the reactions forming X and Y was supposed to leave the system as CO\u003csub\u003e2\u003c/sub\u003e. We wanted to examine whether there could be coexistence of the three cell types. We first just studied the control case (i.e. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\partial\\)\u003c/span\u003e\u003c/span\u003e=1 and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\epsilon 1=\\epsilon ,\\)\u003c/span\u003e\u003c/span\u003e equations below) in which cell type 3 was the same as cell type 1, i.e. with regulation to cell type 2 and responsive to cell type 2 regulation. Then, we considered three \u0026lsquo;asocial\u0026rsquo; cases, i.e. case \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e (\u0026lsquo;autistic and egoistic\u0026rsquo;, i.e. cell type 3 neither regulating/stimulating cell type 2, nor responsive to cell type 2 regulation, i.e. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\partial\\)\u003c/span\u003e\u003c/span\u003e=0 and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\epsilon 1=0\\)\u003c/span\u003e\u003c/span\u003e), case 2 (\u0026lsquo;autistic\u0026rsquo;, i.e. cell type 3 not regulated by cell type 2, cell type 2 still regulated by cell type 3), i.e., \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\partial\\)\u003c/span\u003e\u003c/span\u003e=1 and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\epsilon 1=0\\)\u003c/span\u003e\u003c/span\u003e, respectively. We also considered the remaining case 3 (\u0026lsquo;egoistic\u0026rsquo;, i.e. cell type 3 not regulating cell type 2, but still regulated by cell type 2, i.e. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\partial\\)\u003c/span\u003e\u003c/span\u003e=0 and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\epsilon 1=\\epsilon\\)\u003c/span\u003e\u003c/span\u003e). With the common goods balance met and the glucose influx fixed to 1, the three maximal fluxes towards biomass amounted to:\u003c/p\u003e\u003cp\u003e\u003cimg 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\" width=\"619\" height=\"181\"\u003e\u003c/p\u003e\u003cp\u003eHere \u0026lsquo;\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\epsilon\\)\u003c/span\u003e\u003c/span\u003e\u0026rsquo; or \u0026lsquo;\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\epsilon 1\\)\u003c/span\u003e\u003c/span\u003e\u0026rsquo; is the inter-regulation between cell type 1 or cell type 3 with type 2 in this three-cell-types\u0026rsquo; stepwise growth. Unless specified otherwise, the inherent specific growth rate for cell types 1 and 3 were the same (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\mu }_{1}={\\mu }_{3}=0.2\\)\u003c/span\u003e\u003c/span\u003e), while that of cell type 2 was higher (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\mu }_{2}=0.3\\)\u003c/span\u003e\u003c/span\u003e).\u003c/p\u003e\u003cp\u003eAs per the numerators of the above equations, we tried various values of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\epsilon\\)\u003c/span\u003e\u003c/span\u003e and used the above equations to set the upper bound for the biomass synthesis rate for the three cell types. The sum of the biomass synthesis rates of the three cell types was again the objective function. After defining the model, we calculated the relationship between the cell number and time by using COBRA at every time step and then integrating over time by the above equations for the Biomasses. Finally, we selected case \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e (both autistic and egoistic) for further analysis because that case cell type 3 was most similar in behavior to tumor cells.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec17\" class=\"Section2\"\u003e\u003ch2\u003e4\u0026thinsp;\u0026minus;\u0026thinsp;2 Three cell types regulating each other\u003c/h2\u003e\u003cp\u003eWe could not find a coexistence for three cell types when we chose various \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\epsilon\\)\u003c/span\u003e\u003c/span\u003e values, even though cell type 3 needed support from the other two in terms of common goods X and Y. As cell type 3 modelled a tumor cell, we considered it unlikely that it had completely lost all regulation. Therefore, we next considered the regulatory interactions among three cell types, whereby cell types 1 and 2 mutually regulate each other with strength (elasticity\u003csup\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan class=\"CitationRef\"\u003e33\u003c/span\u003e\u003c/span\u003e,\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e\u003cspan class=\"CitationRef\"\u003e57\u003c/span\u003e\u003c/span\u003e\u003c/sup\u003e) \u0026epsilon;, and both cross-regulate with cell type 3, and \u003cem\u003evice versa\u003c/em\u003e, with a regulation strength (elasticity) 𝛾. The regulated equation is shown below.\u003c/p\u003e\u003cdiv id=\"Equ20\" class=\"Equation\"\u003e\u003cdiv class=\"mathdisplay\" id=\"FileID_Equ20\" name=\"EquationSource\"\u003e\u003cimg 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\" width=\"610\" height=\"83\"\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ21\" class=\"Equation\"\u003e\u003cdiv class=\"mathdisplay\" id=\"FileID_Equ21\" name=\"EquationSource\"\u003e$${b}_{1,s3,r1,ub}\\left(t\\right)=\\frac{{0.5\\bullet \\mu }_{1}\\bullet {{B}_{1,s3,r1}\\left(t-ts\\right)\\bullet \\left({B}_{2,s3,r1}\\left(t-ts\\right)\\right)}^{\\epsilon }{\\bullet \\left({B}_{3,s3,r1}\\left(t-ts\\right)\\right)}^{\\gamma }}{FR1}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e26\u003c/div\u003e\n \u003c/div\u003e\n \u003cdiv id=\"Equ22\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ22\" name=\"EquationSource\"\u003e$${b}_{2,s3,r1,ub}\\left(t\\right)=\\frac{{0.5\\bullet \\mu }_{2}\\bullet {\\left({B}_{1,s3,r1}\\left(t-ts\\right)\\right)}^{\\epsilon }{\\bullet {B}_{2,s3,r1}\\left(t-ts\\right)\\bullet \\left({B}_{3,s3,r1}\\left(t-ts\\right)\\right)}^{\\gamma }}{FR1}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e27\u003c/div\u003e\n \u003c/div\u003e\n \u003cdiv id=\"Equ23\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ23\" name=\"EquationSource\"\u003e$${b}_{3,s3,r1,ub}\\left(t\\right)=\\frac{0.5\\bullet {\\mu }_{3}{\\bullet \\left({B}_{1,s3,r1}\\left(t-ts\\right)\\right)}^{\\gamma }{\\bullet \\left({B}_{2,s3,r1}\\left(t-ts\\right)\\right)}^{\\gamma }\\bullet {B}_{3,s3,r1}\\left(t-ts\\right)}{FR1}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e28\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003ewhere \u0026lsquo;\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\gamma\\)\u003c/span\u003e\u003c/span\u003e\u0026rsquo; is the regulation power (elasticity) between the normal cells and cell type 3, and \u003cem\u003evice versa\u003c/em\u003e. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({b}_{1,s3,r1,ub}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({b}_{2,s3,r1,ub}\\)\u003c/span\u003e\u003c/span\u003eand \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({b}_{3,s3,r1,ub}\\)\u003c/span\u003e\u003c/span\u003e are again used to set the upper bounds for the biomass synthesis reactions. After that, we get the actual biomass synthesis rate for cell types 1, 2 and 3 in this three-cell-types system with regulation by dcFBA for each time point with the sum of their biomass synthesis rates as objective function, then these values will be used for cell number calculation.\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003e4\u0026thinsp;\u0026minus;\u0026thinsp;3 Only cell type 3 (transformed cell) regulated by the two other cell types (normal cells)\u003c/strong\u003e\u003c/p\u003e\n \u003cp\u003eIn actual situations, the transformed cell needs support from normal cells, but itself may not support the normal cells in any way. Accordingly, we constructed another regulation network through equations in which the normal cells regulated the transformed cells, and not \u003cem\u003evice versa\u003c/em\u003e:\u003c/p\u003e\n \u003cdiv id=\"Equa\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equa\" name=\"EquationSource\"\u003e$$FR2={\\mu }_{1}\\bullet {{B}_{1,s3,r1}\\left(t-ts\\right)\\bullet \\left({B}_{2,s3,r1}\\left(t-ts\\right)\\right)}^{\\epsilon }+{\\mu }_{2}\\bullet {\\left({B}_{1,s3,r1}\\left(t-ts\\right)\\right)}^{\\epsilon }\\bullet {B}_{2,s3,r1}\\left(t-ts\\right)$$\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ24\" class=\"Equation\"\u003e\u003cdiv class=\"mathdisplay\" id=\"FileID_Equ24\" name=\"EquationSource\"\u003e$$+ {\\mu }_{3}{\\bullet \\left({B}_{1,s3,r1}\\left(t-ts\\right)\\right)}^{\\gamma }{\\bullet \\left({B}_{2,s3,r1}\\left(t-ts\\right)\\right)}^{\\gamma }\\bullet {B}_{3,s3,r1}\\left(t-ts\\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e29\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ25\" class=\"Equation\"\u003e\u003cdiv class=\"mathdisplay\" id=\"FileID_Equ25\" name=\"EquationSource\"\u003e$${b}_{1,s3,r2,ub}\\left(t\\right)=\\frac{{0.5\\bullet \\mu }_{1}\\bullet {{B}_{1,s3,r1}\\left(t-ts\\right)\\bullet \\left({B}_{2,s3,r1}\\left(t-ts\\right)\\right)}^{\\epsilon }}{FR2}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e30\u003c/div\u003e\n \u003c/div\u003e\n \u003cdiv id=\"Equ26\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ26\" name=\"EquationSource\"\u003e$${b}_{2,s3,r2,ub}\\left(t\\right)=\\frac{{0.5\\bullet \\mu }_{2}\\bullet {\\left({B}_{1,s3,r1}\\left(t-ts\\right)\\right)}^{\\epsilon }\\bullet {B}_{2,s3,r1}\\left(t-ts\\right)}{FR2}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e31\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ27\" class=\"Equation\"\u003e\u003cdiv class=\"mathdisplay\" id=\"FileID_Equ27\" name=\"EquationSource\"\u003e$${b}_{3,s3,r2,ub}\\left(t\\right)=\\frac{{0.5\\bullet \\mu }_{3}{\\bullet \\left({B}_{1,s3,r1}\\left(t-ts\\right)\\right)}^{\\gamma }{\\bullet \\left({B}_{2,s3,r1}\\left(t-ts\\right)\\right)}^{\\gamma }\\bullet {B}_{3,s3,r1}\\left(t-ts\\right)}{FR2}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e32\u003c/div\u003e\u003c/div\u003e\u003cp\u003eAgain, we used equation above to set the upper bound for the biomass production reaction and performed the dcFBA calculation with the sum of their biomass synthesis rates of three cell types as objective function. And we used the biomass synthesis value from dcFBA to calculate the cell number like above.\u003c/p\u003e\u003c/div\u003e"},{"header":"Declarations","content":"\u003cp\u003eDATA AVAILABILITY\u003c/p\u003e\n\u003cp\u003eAll data used for the simulations are presented in this manuscript and its Supplementary files.\u003c/p\u003e\n\u003cp\u003eCODE AVAILABILITY \u0026nbsp;\u003c/p\u003e\n\u003cp\u003eThe code of simulated models is available at Github: https://github.com/YanhuaLiu1/Cells-Competiition\u003c/p\u003e\n\u003cp\u003eAuthor contributions\u003c/p\u003e\n\u003cp\u003eYanhua Liu devised the study, collected literature data, prepared the models, performed all computations, showed and discussed the results and drafted the manuscript. Both authors critically reviewed the data and the manuscript. \u0026nbsp; Hans V. Westerhoff devised the study, discussed the results and edited the manuscript. All authors read and approved the final manuscript.\u003c/p\u003e\n\u003cp\u003eAcknowledgements\u003c/p\u003e\n\u003cp\u003eThis study was supported by a personal development grant from the Guangzhou Elite Project (GEP) Foundation to Yanhua Liu for which we express our gratitude.\u0026nbsp;The funder played no role in study design, data collection, analysis and interpretation of data, or the writing of this manuscript.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eCompeting interests\u003c/p\u003e\n\u003cp\u003eHans V. Westerhoff and Yanhua Liu declare that they have no conflict of interest.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eRocha, B., et al. 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(1976).\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"npj-systems-biology-and-applications","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"npjsba","sideBox":"Learn more about [npj Systems Biology and Applications](http://www.nature.com/npjsba/)","snPcode":"41540","submissionUrl":"https://submission.springernature.com/new-submission/41540/3","title":"npj Systems Biology and Applications","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"NPJ","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"Cell-cell interaction, tumorigenesis, systems biology, flux balance analysis, multicellularity","lastPublishedDoi":"10.21203/rs.3.rs-3059897/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3059897/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eIn multicellular organisms, different cell types compete for resources or growth factors, endangering cellular diversity as well as co-existence. To address this, we developed \u0026lsquo;dynamic cell-cell competition FBA\u0026rsquo; (dcFBA). With total biomass synthesis as objective, we found that lower-growth-yield cell types face extinction even when they synthesized mutually required metabolic commodities. Signal transduction between cells promoted co-existence, when turning the cells into mutually regulatory and responsive \u0026lsquo;social cells\u0026rsquo;. Mutants with specific growth rate but intact signal transduction did not outgrow others. However, loss of its social characteristics enabled a mutant to dominate the other cell types with higher specific growth rates and bring those to extinction. A corollary is that cancer arises from reduced sensitivity to regulatory factors rather than enhanced specific growth rates.\u003c/p\u003e \u003cp\u003eTherapies reinforcing cells\u0026rsquo; cross-regulation, perhaps through alternative signaling routes, may therefore be more effective than those targeting replication rates.\u003c/p\u003e","manuscriptTitle":"‘Social’ versus ‘Asocial’ cells--- Dynamic Competition Flux Balance Analysis","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2023-06-26 18:28:23","doi":"10.21203/rs.3.rs-3059897/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"revise","date":"2023-07-14T05:40:07+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"This content is not available.","date":"2023-07-13T17:52:32+00:00","index":2,"fulltext":"This content is not available."},{"type":"editorInvitedReview","content":"This content is not available.","date":"2023-07-03T20:00:40+00:00","index":1,"fulltext":"This content is not available."},{"type":"reviewerAgreed","content":"This content is not available.","date":"2023-06-21T14:47:18+00:00","index":2,"fulltext":"This content is not available."},{"type":"reviewerAgreed","content":"This content is not available.","date":"2023-06-19T18:59:40+00:00","index":1,"fulltext":"This content is not available."},{"type":"reviewersInvited","content":"","date":"2023-06-18T22:25:32+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2023-06-15T11:39:01+00:00","index":"","fulltext":""},{"type":"submitted","content":"npj Systems Biology and Applications","date":"2023-06-14T15:38:21+00:00","index":"","fulltext":""},{"type":"checksFailed","content":"","date":"2023-06-14T13:30:10+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2023-06-13T20:01:55+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"npj-systems-biology-and-applications","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"npjsba","sideBox":"Learn more about [npj Systems Biology and Applications](http://www.nature.com/npjsba/)","snPcode":"41540","submissionUrl":"https://submission.springernature.com/new-submission/41540/3","title":"npj Systems Biology and Applications","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"NPJ","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"45cf2d97-dc5b-4c81-a02d-4c13755fa9b9","owner":[],"postedDate":"June 26th, 2023","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[{"id":22507528,"name":"Biological sciences/Systems biology"},{"id":22507529,"name":"Biological sciences/Computational biology and bioinformatics"}],"tags":[],"updatedAt":"2023-10-29T07:13:37+00:00","versionOfRecord":{"articleIdentity":"rs-3059897","link":"https://doi.org/10.1038/s41540-023-00313-5","journal":{"identity":"npj-systems-biology-and-applications","isVorOnly":false,"title":"npj Systems Biology and Applications"},"publishedOn":"2023-10-28 04:00:00","publishedOnDateReadable":"October 28th, 2023"},"versionCreatedAt":"2023-06-26 18:28:23","video":"","vorDoi":"10.1038/s41540-023-00313-5","vorDoiUrl":"https://doi.org/10.1038/s41540-023-00313-5","workflowStages":[]},"version":"v1","identity":"rs-3059897","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-3059897","identity":"rs-3059897","version":["v1"]},"buildId":"7rjqhiLT3MXkJMwkYKINL","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}