Self-consistent analytical solutions to the kinetics of lipid-induced protein aggregation

Self-consistent analytical solutions to the kinetics of lipid-induced protein aggregation | bioRxiv /* */ /* */ <!-- <!-- /*! * yepnope1.5.4 * (c) WTFPL, GPLv2 */ (function(a,b,c){function d(a){return"[object Function]"==o.call(a)}function e(a){return"string"==typeof a}function f(){}function g(a){return!a||"loaded"==a||"complete"==a||"uninitialized"==a}function h(){var a=p.shift();q=1,a?a.t?m(function(){("c"==a.t?B.injectCss:B.injectJs)(a.s,0,a.a,a.x,a.e,1)},0):(a(),h()):q=0}function i(a,c,d,e,f,i,j){function k(b){if(!o&&g(l.readyState)&&(u.r=o=1,!q&&h(),l.onload=l.onreadystatechange=null,b)){"img"!=a&&m(function(){t.removeChild(l)},50);for(var d in y[c])y[c].hasOwnProperty(d)&&y[c][d].onload()}}var j=j||B.errorTimeout,l=b.createElement(a),o=0,r=0,u={t:d,s:c,e:f,a:i,x:j};1===y[c]&&(r=1,y[c]=[]),"object"==a?l.data=c:(l.src=c,l.type=a),l.width=l.height="0",l.onerror=l.onload=l.onreadystatechange=function(){k.call(this,r)},p.splice(e,0,u),"img"!=a&&(r||2===y[c]?(t.insertBefore(l,s?null:n),m(k,j)):y[c].push(l))}function j(a,b,c,d,f){return q=0,b=b||"j",e(a)?i("c"==b?v:u,a,b,this.i++,c,d,f):(p.splice(this.i++,0,a),1==p.length&&h()),this}function k(){var a=B;return a.loader={load:j,i:0},a}var l=b.documentElement,m=a.setTimeout,n=b.getElementsByTagName("script")[0],o={}.toString,p=[],q=0,r="MozAppearance"in l.style,s=r&&!!b.createRange().compareNode,t=s?l:n.parentNode,l=a.opera&&"[object Opera]"==o.call(a.opera),l=!!b.attachEvent&&!l,u=r?"object":l?"script":"img",v=l?"script":u,w=Array.isArray||function(a){return"[object Array]"==o.call(a)},x=[],y={},z={timeout:function(a,b){return b.length&&(a.timeout=b[0]),a}},A,B;B=function(a){function b(a){var a=a.split("!"),b=x.length,c=a.pop(),d=a.length,c={url:c,origUrl:c,prefixes:a},e,f,g;for(f=0;f<d;f++)g=a[f].split("="),(e=z[g.shift()])&&(c=e(c,g));for(f=0;f<b;f++)c=x[f](c);return c}function g(a,e,f,g,h){var i=b(a),j=i.autoCallback;i.url.split(".").pop().split("?").shift(),i.bypass||(e&&(e=d(e)?e:e[a]||e[g]||e[a.split("/").pop().split("?")[0]]),i.instead?i.instead(a,e,f,g,h):(y[i.url]?i.noexec=!0:y[i.url]=1,f.load(i.url,i.forceCSS||!i.forceJS&&"css"==i.url.split(".").pop().split("?").shift()?"c":c,i.noexec,i.attrs,i.timeout),(d(e)||d(j))&&f.load(function(){k(),e&&e(i.origUrl,h,g),j&&j(i.origUrl,h,g),y[i.url]=2})))}function h(a,b){function c(a,c){if(a){if(e(a))c||(j=function(){var a=[].slice.call(arguments);k.apply(this,a),l()}),g(a,j,b,0,h);else if(Object(a)===a)for(n in m=function(){var b=0,c;for(c in a)a.hasOwnProperty(c)&&b++;return b}(),a)a.hasOwnProperty(n)&&(!c&&!--m&&(d(j)?j=function(){var a=[].slice.call(arguments);k.apply(this,a),l()}:j[n]=function(a){return function(){var b=[].slice.call(arguments);a&&a.apply(this,b),l()}}(k[n])),g(a[n],j,b,n,h))}else!c&&l()}var h=!!a.test,i=a.load||a.both,j=a.callback||f,k=j,l=a.complete||f,m,n;c(h?a.yep:a.nope,!!i),i&&c(i)}var i,j,l=this.yepnope.loader;if(e(a))g(a,0,l,0);else if(w(a))for(i=0;i (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];var j=d.createElement(s);var dl=l!='dataLayer'?'&l='+l:'';j.src='//www.googletagmanager.com/gtm.js?id='+i+dl;j.type='text/javascript';j.async=true;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-M677548'); Skip to main content Home About Submit ALERTS / RSS Search for this keyword Advanced Search New Results Self-consistent analytical solutions to the kinetics of lipid-induced protein aggregation View ORCID Profile Alisdair Stevenson , View ORCID Profile David Voderholzer , View ORCID Profile Thomas C. T. Michaels doi: https://doi.org/10.1101/2025.07.10.664133 Alisdair Stevenson 1 Department of Biology, Institute of Biochemistry , ETH Zurich, Otto Stern Weg 3, 8093, Zurich, Switzerland 2 Bringing Materials to Life Initiative , ETH Zurich, Switzerland Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Alisdair Stevenson David Voderholzer 1 Department of Biology, Institute of Biochemistry , ETH Zurich, Otto Stern Weg 3, 8093, Zurich, Switzerland Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for David Voderholzer Thomas C. T. Michaels 1 Department of Biology, Institute of Biochemistry , ETH Zurich, Otto Stern Weg 3, 8093, Zurich, Switzerland 2 Bringing Materials to Life Initiative , ETH Zurich, Switzerland Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Thomas C. T. Michaels For correspondence: thomas.michaels{at}bc.biol.ethz.ch Abstract Full Text Info/History Metrics Preview PDF Abstract The aggregation of proteins into amyloid fibrils is a hallmark of several neurodegenerative disorders, including Parkinson’s disease. A growing body of experimental evidence highlights the significant role lipid membranes play in modulating this aggregation process, particularly for proteins such as α -synuclein. Despite this, there has been a lack of quantitative theoretical frameworks capable of describing the kinetics of lipid-induced protein aggregation. In this work, we develop an analytical model that explicitly incorporates lipid-mediated interactions into the aggregation kinetics. By formulating rate equations in terms of lipid surface coverage and applying a fixed-point analysis, we derive self-consistent solutions for the full timecourse of aggregation. Our model captures both one-step and two-step nucleation mechanisms and enables the prediction of key kinetic observables, including half-times and maximal growth rates. These results provide a quantitative foundation for interpreting experimental data and offer new mechanistic insights into how lipids influence the self-assembly of amyloidogenic proteins. I. INTRODUCTION Polymerization of proteins and peptides into filamentous aggregates represents a fundamental form of self-assembly that is crucial for the functioning of biological systems, such as in the case of actin biofilaments 1 . However, aberrant fil-amentous protein aggregation forming amyloid fibrils is associated with severe clinical disorders such as Alzheimer’s and Parkinson’s disease 2 – 4 . Furthermore, due to their unique physicochemical properties, protein filaments are increasingly utilized as biomaterials in nanotechnology 5 , 6 . Given these diverse factors, the field of filamentous protein self-assembly has seen significant activity in recent years, and many efforts have been made to elucidate the underlying mechanisms of protein aggregation. A fundamental question in the field of filamentous protein assembly has been to elucidate the microscopic mechanisms of the aggregation reaction. This information is crucial for the rational design of therapeutic strategies against protein aggregation disorders that target the aggregation process. Historically, mechanisms for simple chemical reactions have been studied through chemical kinetics by formulating rate laws and deriving analytical solutions, which link macroscopic reaction behavior to microscopic processes through quantitative comparison with experimental data. The application of these principles to the formation of protein filaments has yielded substantial insights, particularly through mathematical modeling approaches pioneered by Oosawa, who developed a framework for primary nucleation-driven aggregation 7 . This work was later extended by Eaton and Ferrone to include secondary pathways such as fragmentation and secondary nucleation 8 , 9 . More recent developments have provided analytical solutions to the complete aggregation kinetics with secondary pathways, representing a key step forward in linking theoretical models with experimental data, allowing quantitative comparison of theoretical predictions with experimental data describing protein aggregation kinetics 10 – 14 . However, one limitation of these chemical kinetics models of aggregation is that they typically assume spatial homogeneity, neglecting the impact of local environmental factors on aggregation behavior. A particularly relevant example of a spatial heterogeneity that modulates aggregation is lipid surfaces, such as in the case of α -synuclein aggregation, a process associated with Parkinson’s disease. Here, the N-terminal region and part of the non-amyloid core of α -synuclein can bind to lipid surfaces and these interactions may form part of the in vivo function and pathology of α -synuclein 15 – 27 . Notably, lipid interactions have been shown to modulate the aggregation kinetics of α -synuclein, potentially altering nucleation and fibril growth pathways 23 – 44 . Here, we describe a kinetic framework that explicitly incorporates lipid-induced modulation of protein aggregation in terms of kinetic equations for the surface coverage of monomers and aggregates and derive analytical self-consistent solutions to these rate equations that describe the full timecourse of aggregation in terms of the underlying rate parameters. We discuss solutions in a number of limits, derive important characteristics of the reaction, including half-times, from first p rinciples. Our results provide a quantitative basis for interpreting experimental data and offer new insights into the impact of lipids on protein self-assembly, and were successfully applied to elucidate the mechanism of lipidinduced α -synuclein aggregation 45 . The paper is structured as follows. In Section II, we introduce the theoretical framework of lipid-induced protein aggregation, developing kinetic equations based on surface coverage variables and considering both one-step and two-step primary nucleation pathways. Section III leverages the separation of timescales between lipid binding and aggregation to simplify the kinetic model and derive a reduced system within a slow manifold. In Section IV, we outline our fixed-point iterative approach for obtaining self-consistent analytical solutions, with Section V providing linearized solutions that serve as the initial approximation. Section VI presents first-order self-consistent solutions, while Section VII characterizes key features of the aggregation kinetics, including early-time behavior, steady-state yields, maximal growth rates, and half-time scaling laws. In Section VIII, we extend our model to account for lipid-limited conditions at low lipid-to-protein ratios. II. THEORY OF LIPID-INDUCED PROTEIN AGGREGATION KINETICS We begin by introducing the kinetic equations for lipid-induced protein aggregation. We consider surface coverages of protein monomers and aggregates as the key quantities in our model, rather than their absolute concentrations ( Fig. 1a ). These surface coverages represent the fraction of the lipid surface covered by protein monomers or aggregates, taking values between 0 (empty surface) and 1 (fully covered surface), Fig. 1a . The protein monomer coverage θ m is defined through the equation Download figure Open in new tab FIG. 1. (a) Schematic representation of protein monomer surface coverage of a lipid surface. For simplicity, each protein monomer consumes one lipid binding site in this illustration ( β = 1), but our theoretical framework can account for variable stoichiometries of protein-lipid binding through the β parameter. (b) Schematic representation of the elementary microscopic mechanisms underlying the formation and propagation of lipid-protein coaggregates considered in our model. These reaction mechanisms are both protein and lipid-dependent, where the variable contribution of bound and free protein monomers to coaggregate nucleation is described by the reaction orders n 1 and n 2 . For elongation, bound, not free, protein monomers are required as the lipids are also incorporated into the coaggregate structure. where m b is the concentration of bound monomer, L tot is the total lipid concentration (surface sites) and β is the average number of lipids that bind to a protein monomer in monomeric form. Similarly, we can introduce surface coverages for aggregates as Here, we distinguish between fibril mass concentration M and fibril number concentration P . Consequently, θ P is the probability of meeting a fibril end on the surface, produced by normalizing the number concentration of fibrils by the concentration of lipids. α is the average number of lipids that bind to a protein monomer in fibril form. A. Binding kinetics We first consider the binding-dissociation kinetics of protein monomers to the lipid surface. To derive a dynamic equation in terms of protein monomer surface coverage, we describe the dynamics in terms of bulk concentrations of free and bound protein monomers to give where m b and m f are the bound and free concentrations of protein monomer respectively, and L is the concentration of free lipid sites. The rate constants k on and k off describe the rates of protein-lipid binding and dissociation respectively ( Fig. 1b ). The bound protein concentration can be converted to surface coverage using Eq. (1a) . The free protein monomer concentration can be described implicitly in terms of surface coverage terms by considering the conservation of protein mass in the system to give where the surface coverage of oligomers is neglected as θ S ≪ θ m at early-times and θ S ≪ θ M at late-times in the reaction. The fraction of free lipid sites is θ L = 1 − θ m − θ M (again neglecting θ S ), which follows from the conservation of lipid surface sites in the closed system, and can be converted to describe L : Taken together, we obtain the differential equation describing the dynamics of protein monomer surface coverage due to protein-lipid binding and dissociation as B. One-step primary nucleation In addition to the binding flux of protein monomers to the lipid surface, we consider fibril nucleation and growth events occurring in contact with the surface that produce lipid-protein coaggregates 37 , where these microscopic aggregation processes are shown graphically in Fig. 1b . In the simplest scenario, classical nucleation theory describes the nucleation process as occurring in one step, described by The first and second terms of Eq. (6a) describe the binding flux of protein monomers to the lipid surface, as described in Eq. (5) . The third term of Eq. (6a) is the reaction flux for coaggregate elongation, where k + is the rate of coaggregate elongation. This expression arises because of the assumption that coaggregate elongation is possible only via the incorporation of both protein monomers and lipids localized to the lipid surface. Therefore, the fibril elongation rate is proportional to θ m , rather than the concentration of protein monomers. The factor of 2 accounts for the possibility of adding monomers at either end of the fibrils. Eq. (6c) is the reaction flux of one-step primary nucleation, where k n is the associated rate constant. We allow for the different possible contributions of free and bound protein monomers to the primary nucleation by using variable reaction orders n 1 and n 2 respectively. As fibril nucleation does not necessarily occur only on the lipid surface, immediately creating lipid-protein coaggregates, we convert the concentration to surface coverage with θ P = P / L tot . Note that in Eq. (6a) and (6b) , we have neglected the contribution of nucleation to the dynamics of θ m and θ M . This is justified as the reaction flux due to filament elongation is significantly larger than the flux due to primary nucleation, which ensures the formation of sufficiently elongated structures. Consequently, elongation is the dominant reaction flux in the conversion process from proteins in monomeric to fibril form, rendering the dynamics of nucleation negligible for the dynamics of θ m and θ M . C. Two-step primary nucleation We also consider that the nucleation of filamentous structures often proceeds through a multi-stage process rather than a single step as proposed by classical nucleation theory. This non-classical nucleation pathway includes intermediate oligomeric states, where in our study, we examine a two-step primary nucleation mechanism where protein monomers first nucleate to form a generic, on-pathway, oligomer, that then converts to an elongation-capable coaggregate ( Fig. 1b ). A two-step primary nucleation mechanism has been identified in numerous amyloid-forming systems 46 – 48 , as well as in other nucleation processes, including crystallization 14 , 49 – 51 . To describe this scenario, we extend Eq. (6) to include the concentration of intermediate oligomers and additional microscopic mechanisms to give where θ S is the concentration of oligomers and k o and k c are the rate constants for the primary nucleation of oligomers and conversion of oligomers to coaggregates respectively. Eq. (7a) and (7b) remain unchanged compared to the one-step model, Eq. (6) . The first term in Eq. (7c) describes oligomer formation, with variable contributions of bound and free protein monomers. We implement a conversion process of oligomers to coaggregates with a linear dependence on the oligomer concentration, and no dependence on bound or free protein monomers, resulting in the oligomer conversion flux k c θ S , where k c is the rate of conversion of oligomers to coaggregates. In principle, our model could describe a protein monomer dependent conversion process. In Eq. (7) , we assume that the process of oligomers binding to the lipid surface is not rate-limiting and thus do not explicitly describe it. Additionally, oligomer dissociation into constituent monomers is also disregarded as we do not have direct experimental measurements of oligomer concentration. In principle, our model can be extended to take both processes into account. If oligomer dissociation is explicitly described, the timescale for oligomer conversion is determined by an effective rate dependent on the rates of oligomer dissociation and conversion k e = k d + k c , thereby defining the steady-state timescale for oligomer concentration. III. PRE-EQUILIBRIUM OF PROTEIN TO LIPID BINDING The first step to determine self-consistent solutions to the aggregation kinetics described by Eqs. (6) and (7) is to recognize the separation of timescales between the binding dynamics and aggregation reaction. Typically, the protein-lipid binding rate is much faster (on the scale of minutes) than the characteristic timescale of aggregation, which is the time for half the protein monomer mass to aggregate (on the scale of several hours), which has been validated experimentally 44 . With a separation of timescales between these processes, the kinetic models can be analyzed utilizing mathematical techniques based on matched asymptotics 52 and allow for the division of the overall dynamics of the system into two stages: Initial layer – an initial rapid phase where aggregation is minimal, and a pre-equilibrium is established for protein-lipid binding. Slow manifold – a slower phase where aggregates form and grow, with the binding dynamics robustly maintaining a pre-equilibrium throughout. Since the binding dynamics is much faster than the subsequent aggregation, we can neglect the initial layer and focus on the dynamics within the slow manifold when developing self-consistent analytical solutions to the aggregation kinetics (see Fig. 2 for a comparison of timescales). Here, the binding pre-equilibrium simplifies the dynamics to a lowerdimensional system, by setting the binding and unbinding terms in Eqs. (6a) and (7a) to zero due to the rapid preequilibrium dynamics, establishing a relationship between θ m and θ M , described by Download figure Open in new tab FIG. 2. The separation of timescales between the kinetics of binding-dissociation of protein monomers to the lipid surface and subsequent lipid-induced protein aggregation reaction kinetics. Parameters used for this plot: L tot = 100 µM, m tot = 25 µM, n 1 = n 2 = 1, β = 28.2, α = 11.4, K D = 3.8 × 10 −1 µM −1 , . Download figure Open in new tab FIG. 3. Comparison of the exact solution describing the dynamics of normalized fibril mass concentration for the two-step primary nucleation model derived from numerically evaluating the ODE system to the linearized analytical solution, M (0) ( t ), and self-consistent solution of order one, M (1) ( t ). Parameters used for this plot: n 1 = 0, n 2 = 1.5, k + k o = 0.0118 hr −2 µM −1 , k c = 0.265 hr −1 and otherwise are identical to those used in Fig. 2 . This equation can be solved for θ m as a function of θ M to yield where is the dissociation constant of protein-lipid binding. To make analytical progress, we linearize Eq. (9) on θ M to yield the simplified expression Where is the monomer surface coverage when θ M = 0 and Where is the lipid-to-monomer ratio. The dynamics in the slow manifold are of a lower dimension due to the constraint, defined by Eq. (10) , where θ m can be described in terms of θ M and reduce the number of independent kinetic equations. In particular, implementing the linearized binding pre-equilibirum, Eq. (10) , into the kinetic equations of the one-step primary nucleation model, Eq. (6) , produces a reduced set of dynamic equations in the slow manifold Similarly, implementing Eq. (10) into the two-step primary nucleation model, Eq. (7) , produces Therefore, by exploiting the separation of timescales between the binding dynamics and aggregation, we have reduced the system of kinetic equations to describe the dynamics in the slow manifold. In the following sections, we outline strategies based on fixed-point iterations to obtain accurate selfconsistent solutions to these kinetic equations in the slow manifold. IV. SELF-CONSISTENT SOLUTIONS FOR LIPID-INDUCED AGGREGATION The methodology for developing analytical solutions for Eqs. (14) and (15) is to utilize fixed-point iterations. 10 , 11 This approach allows self-consistent solutions with increasing accuracy to be derived in an iterative process. Self-consistent techniques have been previously applied to derive analytical solutions for the kinetics of protein filament formation in the absence of lipids. 10 , 11 To transform the differential equation systems (14) and (15) into a fixed-point problem, we integrate the kinetic equations such that they are of the form for some operator 𝒜 with for the onestep primary nucleation and for the two-step primary nucleation model. Therefore, the desired solution to the system of differential equations (Eqs. (14) and (15)) is the fixed point of the operator 𝒜. This problem is solved self-consistently if a fixed point of the operator 𝒜 satisfying is found. According to the contraction mapping principle, for an initial guess sufficiently close to the fixed point , the fixed point can be computed iteratively as The convergence of the fixed-point iteration depends on the choice of the initial guess. Therefore, in the following section, we choose as an appropriate starting point for the iteration procedure the early-time linearized solution to the kinetic equations. This choice produces highly accurate selfconsistent solutions for the full aggregation timecourse even after one iteration step. To derive the fixed-point operators for the one-step and twostep models, we reformulate the kinetic equations Eqs. (14) and (15) as fixed-point problems through formal integration. For the one-step primary nucleation model, the fixed-point problem is: where the right-hand side of Eq. (18) is the fixed-point operator 𝒜. Similarly, Eq. (15) can now be rewritten as where the right-hand side of Eq. (19) is the respective fixedpoint operator 𝒜 for the two-step primary nucleation model. Hence, the general form of θ M ( t ) in arbitrary order of the fixed point operation (except for zeroth order) will be Where is a function that depends on the specific kinetic parameters in the model. Interestingly, the solution for at any order in both the one-step and two-step models has this form. In the following sections we will derive explicit expressions for F ( n ) ( t ) through the application of the fixed-point iteration. V. SOLUTIONS TO THE LINEARIZED KINETIC EQUATIONS As the starting point in our fixed-point analysis, we use the linear solutions to Eqs. (14) and (15) that emerge in the earlytime limit, when the surface coverage of protein monomers can be assumed constant at its initial value. Mathematically, this is represented by setting θ m ( t ) equal to the initial value θ m (0) = θ 0 for small times t , where θ 0 is defined in Eq. (11) . The solutions to the linearized kinetic equations capture the aggregation kinetics at early times, when monomer depletion through their incorporation into aggregates is negligible, and thus provide a valuable starting point for the fixed-point iteration method, facilitating its convergence. Below, we discuss the solutions to the linearized kinetic equations for each model (one-step and two-step primary nucleation) separately. A. One-step primary nucleation model Setting θ m ( t ) = θ 0 in Eq. (14) and using θ M ( t ), θ P ( t ) ≪ θ 0 at early-times yields the following linearized kinetic equations The solution for the initial values θ P (0) = θ M (0) = 0 reads: where . B. Two-step primary nucleation model The linearized kinetic equations for the two-step primary nucleation model are: With the initial values θ S (0) = θ P (0) = θ M (0) = 0 the solution is where . VI. SOLUTIONS TO THE NON-LINEAR MOMENT EQUATIONS Accurate self-consistent solutions for the full timecourse of the aggregation reaction can now be obtained iteratively by applying the operator 𝒜 to the early-time solutions obtained in Sec. V. A. One-step primary nucleation model The first-order self-consistent solution for the aggregate mass is obtained by substituting the linearized solution , Eq. (23a) , into the fixed point operator Eq. (18) Performing the integration yields the first-order selfconsistent solution for M ( t ) as Where is an effective rate constant of aggregate proliferation through the combined effects of nucleation and elongation. This effective rate generalises the effective rate constant derived by Oosawa for nucleated polymerisation in the absence of lipid surfaces. B. Two-step primary nucleation model Repeating the analysis for the two-step primary nucleation model, we obtain the first-order self-consistent solution by inserting Eq. (25b) into the fixed-point operator Eq. (19) On integration, Eq. (29) becomes: Where is the effective rate constant for aggregate proliferation, where k n is replaced by k o . With these solutions, the effect of varying lipid and protein monomer concentrations on the aggregation kinetics is summarized in Fig. 4a and b, respectively. Download figure Open in new tab FIG. 4. (a) & (b) Dynamics of normalized fibril mass predicted by the first order self-consistent solution to the two-step primary nucleation model for varying lipid-to-protein ratios, r , where the lipid (a) and protein (b) concentration is varied. (a) The vertical arrow indicates the change in behavior as lipid concentration is increased in the lipid-limited regime ( r α ). (b) The diagonal arrow indicates the change in behavior as protein concentration is decreased across r values in both the lipid-limited and protein-limited regimes. (c) & (d) Dependence on the steadystate yield of lipid-protein coaggregates on the lipid-to-protein ratio, r , where the lipid (c) and protein (d) concentration is varied. Parameters used in this plot: n 1 = 0, n 2 = 1.5, β = 28.2, α = 13, K D = 3.8 × 10 −1 µM −1 , k + k o = 0.0118 hr −2 µM −1 , k c = 0.265 hr −1 . Panel (a): m tot = 20 and L tot = 40, 80, 130, 200, 260, 400, 500, 600 µM . Panel (b): L tot = 100 µM and m tot = 3.33, 4.76, 6.45, 7.69, 10, 15.38, 25, 50 µM . VII. CHARACTERISTICS OF LIPID-INDUCED PROTEIN AGGREGATION KINETICS Having derived analytical expressions for the full timecourse of aggregation, we are now in a position to derive from first principles a series of important characteristics of lipidinduced protein aggregation. These include an analysis of the steady-state behavior as well as a discussion of half-time scaling. A. Steady-state behavior The plateau concentration of aggregates can be found by taking the t → ∞ limit of Eqs. (27) and (30), yielding in both Cases Interestingly, we find a biphasic thermodynamic behavior depending on the lipid-to-monomer ratio, where these solutions correspond to two possible outcomes of the aggregation reaction: M ( t = ∞) = L tot / α or M ( t = ∞) = m tot . In nonlinear dynamics theory, this behavior is defined as a transcritical bi-furcation. When the lipid-to-monomer ratio r is below the critical value α , the final aggregate yield increases proportionally to the lipid concentration. Crucially, in this regime, not all protein monomers are converted into aggregates by the end of the reaction. There remains a surplus of protein monomers. By contrast, for r ≥ α , the final aggregate load is set by the available monomer concentration. In this case, all protein monomers are converted to fibrils, and there is an excess of lipids remaining at the end of the reaction. This biphasic behavior highlights the critical influence of the lipid-to-monomer ratio on the final state of the aggregation process. The steady-state behavior of the system when lipid and protein monomer concentrations are varied is summarized in Fig. 4c and d, respectively. B. Early-time limit In the early-time limit λt ≪ 1, the aggregate mass concentration follows a polynomial increase with time that corresponds to the solution to the linearized form of equation. For the one-step nucleation model, expanding the Eq. (27) for early times yields View this table: View inline View popup Download powerpoint TABLE 1. Comparison of the first-order self-consistent solutions to lipid-induced protein aggregation distinguishing the one-step and two-step nucleation models. In this case, the early-time aggregate mass displays a quadratic dependence on time, M ( t ) ∝ t 2 . This result exhibits the same early-time behavior as the Oosawa model of nucleated polymerization, which also shows a quadratic increase at early times. A similar polynomial early-time behavior is observed for the two-step nucleation model. Expanding in Eq. (30) for k c t ≪ 1 yields Expanding the exponential in Eq. (34) further for early times yields In the two-step nucleation model, the aggregate mass increases in proportion to t 3 , as opposed to the quadratic increase t 2 observed in the one-step nucleation model. This reflects the additional complexity of the two-step nucleation process, where the formation of intermediates leads to a slower initial growth rate. More generally, the introduction of multi-step nucleation pathways can lead to even higher-order polynomial dependencies at early times. When nucleation involves multiple conversion steps before fibril elongation, the early-time kinetics are governed by: where n is the number of rate-limiting nucleation-conversion steps 53 . This means that higher-order nucleation cascades systematically increase the power-law exponent of the initial growth phase. C. Slow and fast conversion limits In the limit of fast oligomer conversion, the argument of the exponential in Eq. (30) becomes and therefore Eq. (30) recovers the first-order self-consistent solution for the one-step nucleation model In the slow oligomer conversion limit, Eq. (30) simplifies to Where is a new effective rate controlling aggregate proliferation in the limit of slow conversion. Note that µ is the geometric average of the three rate constants of oligomer formation, oligomer conversion, and fibril growth. D. Maximal growth rate The maximal growth rate quantifies the fastest rate of aggregate formation and is defined as: where t max is the inflection point, defined by the equation (see Fig.5a). For the one-step nucleation model, using Eq. (27), we find t max = 1/ λ and therefore the maximal growth rate is Similarly, for the two-step nucleation model, we obtain from Eq. (39) that t max = 4 1/3 / µ and thus the maximal growth rate in this case reads Fig. 5b presents a plot of the maximal growth rate σ max , revealing a peak at r = α , which marks the transition between the lipid-limited and protein-limited regimes. Download figure Open in new tab FIG. 5. (a) Dynamics of normalized fibril mass predicted by the first order self-consistent solution to the two-step primary nucleation model detailing the location of maximal growth rate, σ max , and half-time of the aggregation reaction, t 1/2 . Parameters used for this plot: (b) Dependence on the amplitude of the maximal growth rate, σ max , on the lipid-to-protein ratio, r . The maximum of σ max is achieved at the critical ratio r = α where the system switches from being lipid-limited to protein-limited with respect to coaggregate yield. Parameters used for this plot are identical to those used in Fig. 3 . E. Half-time scaling A key measure often used in characterising protein aggregation is the half-time t 1/2 , which is the time taken for the fibril mass concentration to reach half of its plateau value. In previous studies of homogeneous aggregation (i.e. in the absence of lipid surfaces), it has been shown that t 1/2 displays a scaling relationship with the initial protein monomer concentration, .The scaling exponent, γ , depends on the reaction orders of the dominant nucleation mechanism. Therefore, experimental measurements of γ (e.g. through a double logarithmic plot of t 1/2 with m tot ) can reveal important mechanistic information about the underlying aggregation mechanisms. protein monomer concentration, m tot , and the initial lipid concentration, L tot and examine the t 1/2 scaling. Interestingly, we find a distinct t 1/2 scaling when varying m tot or L tot and, for both of these cases, the scaling of t 1/2 exhibits multiphasic behavior with distinct scaling exponents in the lipid and protein limited regimes. For this reason, we introduce two scaling exponents γ L and γ p to capture the dependence of t 1/2 on L tot and m tot , respectively: We now derive explicit expressions for these scaling exponents γ L and γ p valid in the different regimes. 1. Fast conversion limit (one-step nucleation) The half-time of aggregation can be found explicitly by the solving , which using Eq. (27) yields It is convenient to explicitly consider the form of Eq. (45) in the two regimes r < α and r ≥ α : When r < α , we have A = 1 and Eq. (45) reads explicitly When r ≥ α , we have A = α / r and therefore Eq. (45) becomes: To understand how Eqs. (46) and (47) scale in the limit of low/high values of m tot and L tot , four cases were examined. Case 1: m tot = constant & L tot → 0 ( r α ). In this limit, The t ½ scaling can now be found by examining (47): so as L tot → ∞: Case 3: L tot = constant & m tot → 0 ( r > α ). In this limit, and using Eq. (47) in the protein limited regime, we find: In other words Case 4: L tot = constant & m tot → ∞ ( r < α ). In this limit, θ 0 ≈ 1 and therefore from Eq. (46) , we find The scaling exponents can be converted to the form t 1/2 , and the scaling results are summarized in Fig. 6 , where the scaling behavior of the one-step primary nucleation model is identical to the scaling of the two-step primary nucleation model with fast oligomer conversion. Download figure Open in new tab FIG. 6. (a) Schematic log-log plot of t 1/2 against r qualitatively describing the six distinct scaling regimes, contingent upon whether the aggregation reaction is limited by initial lipid or protein concentration. In region 1, the reaction is lipid-limited and r < r * . In region 2, the reaction is lipid-limited but r * < r α . (b) Table summarizing theoretical predictions for the scaling exponents γ L and γ p depending on r . 2. Slow conversion limit (two-step nucleation) In the limit of slow oligomer conversion, we use Eq. (39) to determine the half-time, yielding the following expression: Again, it is useful to distinguish between the two regimes r < α and r ≥ α , respectively By using the same limits for θ 0 as in Section VII E 1, we find the following scaling exponents for t 1/2 . In the regime r α , we find VII. LOW LIPID-TO-MONOMER RATIO: LIPID-TRANSPORT LIMITED REGIME Experimental data indicate that when r falls below a critical threshold ( r * ), kinetic traces slow down as r decreases. This behavior suggests that the elongation process, which is typically lipid-driven, becomes constrained by lipid scarcity, requiring a revision to our theoretical framework. Rather than proceeding as a single-step reaction, lipid-driven elongation transitions to a two-step process at low r . To account for this, we introduce a correction factor that modifies the nucleation and elongation fluxes in our equations. The key adjustment involves correcting θ m in Eq. (6), respectively, Eq. (7) with a term that explicitly accounts for lipid scarcity This factor ensures that, for r r * , the correction term approaches unity, preserving the original form of the model. A. Self-consistent solutions With revised nucleation and elongation fluxes, selfconsistent solutions retain their original structure (Eq. (30)), but incorporate effective scaling terms for λ and µ . Specifically, we introduce the following transformations: These modifications ensure that the lipid limitation is explicitly accounted for in the rate equations, leading to corrected aggregation kinetics in the low- r regime. B. Half-time scaling The introduction of the lipid-dependent correction in λ and µ directly impacts the scaling behavior of the half-time ( Fig. 6 ). Specifically, for r < r * , the scaling exponents γ L and γ p are reduced due to the lipid limitation. For one-step nucleation, the correction modifies the scaling exponent as follows: while for two-step nucleation, the reduction follows a different scaling factor: IX. CONCLUSIONS In this work, we have developed a self-consistent theoretical framework to describe the kinetics of lipid-induced protein aggregation. By explicitly incorporating lipid-dependent nucleation and elongation processes, our model extends classical polymerization theories 7 , 10 to account for the critical role of lipid availability in modulating aggregation behavior. A key advantage of this framework is the availability of analytical solutions, which will enable the interpretation of experimental aggregation data in terms of the underlying rate constants of the microscopic steps of aggregation. By offering a quantitative means to extract kinetic parameters from experimental data, this framework lays the groundwork for future studies exploring lipid-mediated protein aggregation in physiological and pathological settings. Future extensions of this model could incorporate additional processes such as secondary nucleation and aggregation inhibitors to further refine our understanding of lipid-protein interactions in aggregation. DATA AVAILABILITY The data that support the findings of this study are available within the article. ACKNOWLEDGMENTS This work was funded by ETH Zurich (AS, TCTM) and the Swiss National Science Foundation (grant no SNS 219703 to TCTM). Funder Information Declared Swiss National Science Foundation, https://ror.org/00yjd3n13 , SNS 219703 References ↵ T. P. Knowles , T. W. Oppenheim , A. K. Buell , D. Y. Chirgadze , and M. E. 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