A model of protein folding with multiple native states: Metamorphicity, Intrinsic disorderness and folding upon binding of proteins

A model of protein folding with multiple native states: Metamorphicity, Intrinsic disorderness and folding upon binding of proteins | 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 A model of protein folding with multiple native states: Metamorphicity, Intrinsic disorderness and folding upon binding of proteins Rohon Mitra , Biman Jana doi: https://doi.org/10.1101/2025.02.26.640301 Rohon Mitra 1 School of Chemical Sciences, Indian Association for the Cultivation of Science , Kolkata 700032, India Find this author on Google Scholar Find this author on PubMed Search for this author on this site Biman Jana 1 School of Chemical Sciences, Indian Association for the Cultivation of Science , Kolkata 700032, India Find this author on Google Scholar Find this author on PubMed Search for this author on this site For correspondence: pcbj{at}iacs.res.in Abstract Full Text Info/History Metrics Supplementary material Preview PDF Abstract The sequence-structure-function paradigm in biology states that protein’s amino acid sequence determines its unique folded state structure which in turn dictates its unique biological function. This classic concept has been severely challenged by the discovery of metamorphic and intrinsically disorder proteins (IDPs). Metamorphic proteins can fold into multiple native structures and perform multiple functions. IDPs, on the contrary, remain unstructured at physiological conditions, but can fold to unique structure upon binding to a target protein and show functionality. Here, we present a statistical mechanics model of protein folding with multiple native states and analyze their folding phase diagrams. While recovering classic sequence-structure-function paradigm for single native state, our model shows metamorphicity at lower number of native states. An expansion of unfolded region in the phase diagram at higher number of native states, making unfolded state as the stable state at physiological conditions, indicates the emergence of IDP-like scenario. Folding upon binding scenario of IDPs has also been demonstrated when an energetic bias is introduced for a specific native state. A regaining of folded region upon biasing, increasing the folding propensity of the system, with folding towards a specific native state is shown. Therefore, our model is general enough to reproduce the classic sequence-structure-function, metamorphicity, intrinsic disorderness and folding upon binding scenarios of proteins. Significance Statement Proteins exhibit remarkable structural diversity, from having unique single native structure to having no particular structure but an inherently disordered ensemble that usually attains a single structure upon binding to another target. Here, we present a general statistical mechanics model of protein folding that reproduces single native state folding, metamorphicity, intrinsic disorder, and bindinginduced folding by incorporating multiple native states. A notable increase in propensity of being in unfolded state, resembling intrinsic disorder at physiological conditions, for system with larger number of native state was shown. Introduction of a bias was shown to increase the folding propensity of such proteins directed toward a specific native state. The model’s ability to capture a wide range of structural diversity in proteins makes it highly versatile. Introduction Protein folding is an intricate and highly regulated process that involves the transformation of a linear chain of amino acids into a specific, functional three-dimensional structure. The Anfinsen hypothesis [ 1 ] states that the final structure of a protein is determined by its amino acid sequence, meaning that the sequence contains all the information needed for it to fold properly under the right conditions [ 2 , 3 ]. This process is essential for the proper functioning of proteins, as the final folded structure determines the ability of the protein to interact with other molecules and perform its biological functions [ 4 – 8 ]. On the issue of folding timescale, Levinthal’s paradox [ 9 ] points out that randomly sampling all possible conformations would take an impractically long time for a protein to reach its native state. How-ever, despite the many possible conformations, a protein efficiently finds its correct structure within a biologically relevant timescale. Therefore, proteins do not fold through random sampling; rather, the amino acid sequence encodes the necessary information to guide the folding process along specific energetically favorable pathways [ 10 – 12 ], navigate through a complex energy landscape [ 13 – 15 ] that directs the protein towards its stable native state by avoiding high-energy barriers [ 16 , 17 ] and favoring low-energy conformations. The sequence dictates how the protein will fold, and certain regions form secondary structures such as alpha helices and beta sheets, which then assemble in the final tertiary structure [ 18 ]. In recent years, advances in computational approaches, particularly in sequence-to-structure prediction, have made significant progress toward understanding protein folding. One of the most notable breakthroughs is AlphaFold, an AI system that predicts protein structures from amino acid sequences with unprecedented precision [ 19 , 20 ]. However, while AlphaFold successfully predicts native structures, it does not reveal the folding pathway [ 21 , 22 ], the step-by-step process by which the protein achieves its functional form. When the folding process goes wrong, proteins can misfold [ 23 – 25 ] and aggregate [ 26 ], leading to dysfunctional proteins associated with diseases [ 27 ], such as Alzheimer’s [ 28 ] and Parkinson’s [ 29 , 30 ]. Most recently, the concept of one sequence providing one unique structure/function is challenged. Certain proteins, such as metamorphic proteins [ 31 – 33 ], can adopt multiple stable conformations, allowing them to switch between functional states depending on environmental conditions [ 34 , 35 ]. Another important class of proteins, Intrinsically disordered proteins (IDPs) do not even fold into fixed three-dimensional structures but remain flexible, exist in various conformations [ 36 , 37 ], and often play a critical role in signaling and regulation. However, these IDPs can adopt specific three-dimensional structures when bound to specific target proteins and can perform multiple functions [ 38 , 39 ]. To better understand protein folding, various physics-based models have been developed, each capturing different aspects of the folding process. One such foundational model is the spin-glass framework [ 40 – 45 ], originally developed for disordered magnetic systems. In their seminal work, Bryngelson and Wolynes have developed a theoretical model of protein folding with a single native state in a modified spin-glass framework. [ 46 ] Using a mean-field approach with random energy approximation, the model provides a phase diagram for protein with 3 main phases: the unfolded disordered state, the native folded state, and a glassy phase. [ 47 , 48 ] The model reveals that, under certain conditions, the system undergoes a first-order phase transition between folded and unfolded states. Furthermore, the glassy phase suggests the presence of a second-order phase transition [ 49 ] in the system as well. It captures the complexity of rugged energy landscapes with numerous local minima. However, a theoretical framework that includes scenarios like metamorphic proteins, IDPs, and folding upon binding of IDPs is still lacking. In this work, we present a theoretical model to investigate the folding behavior of proteins with multiple native states. We first develop a generalized framework for systems with n native states and analyze the phase diagram features for n = 1 to 5. The results are correlated with metamorphic proteins and intrinsically disordered proteins (IDPs). Moreover, our model naturally captures folding-upon-binding scenarios observed in IDPs. Model We base our approach on the spin-glass model of protein folding, extending it to systems with multiple native states, as first proposed by Wolynes and co-workers [ 46 , 50 ] in their seminal work on protein folding dynamics. In this model, the folding process is governed by a complex energy landscape with multiple local minima, where proteins navigate through various configurations to reach their folded functional state. To describe the total energy of a configuration of a protein, different energy terms were defined: ε i ( α i ) represents the energy of i th amino acid residue in its specific conformation, α i . J i , i +1 ( α i , α i +1 ) accounts for nearest-neighbor interactions between two successive amino acids in the peptide chain, and K i , j ( α i , α j , r i , r j ) describes long-range interactions between residues that are spatially close in the tertiary structure of that configuration of the protein. Here, α i denotes the state of a particular residue at position r i in the sequence, and each residue can adopt one native state along with approximately 10 non-native states( ν ≈ 10), as originally suggested by Wolynes and co-workers. The total energy of the protein in a particular configuration is therefore expressed as: However, working with Eq. (1) analytically is highly challenging due to the complexity introduced by a large number of variables. A stochastic Hamiltonian approach is introduced to statistically model energy states, capturing the essential features of spin-glass behavior. In line with Derrida’s random energy approximation, they assumed that the energies of different protein conformations are uncorrelated, simplifying the model while maintaining its key characteristics. Considering a protein with N residues, where N 0 number of residues are in their native state, there can be different possible protein configurations with different total energies. The joint probability distribution for n configurations with energies E 1 , E 2 , …, E n is given by: At the residue level, non-native configuration energies are considered to be with mean and standard deviation of Δ ε , while all native configuration energies are set at − ε 0 , where . For the nearestneighbor interaction term, non-native interactions have a mean of and standard deviation of Δ J , with all native interactions set to − J , where . At the tertiary level, non-native interactions follow a distribution with mean and standard deviation of Δ K , while all native interactions are set to − K , where . Typically, the number of tertiary interactions for each residue (z) is around 2–3 for protein, with z varying slightly between the completely unfolded and fully folded conformations. The random energy approximation applies to these structural interactions and the distribution of total energy with N 0 number of residues are in their native state is modeled as a Gaussian distribution with the average being: where: The standard deviation of the distributionΔ E ( N 0 ) is given by: where: In our model, while keeping the basic parameters unchanged, we assume that each amino acid residue can exist in more than one native state and ten non-native states. Let us assume a situation where N 0 number of residues are in the native state out of total N residues. The total count of native residues will now be distributed across n alternative native states, with N i residues in the i -th native state. The combined total of residues across all native states satisfies the equation: .The energies of different protein configuration under these condition is a Gaussian probability distribution P ( E, N 1 , N 2 , …, N n ), where N i represents the number of residues in the i -th distinct native state. The average energy of this distribution is given by The standard deviation is expressed as By comparing Eqs. 2 and 3 with Eqs. 5 and 6, it becomes evident that, for a given value of N 0 , the multi-state model exhibits a lower average value of the distribution compared to the single-state model. Additionally, the standard deviation is larger in the multi-state model, reflecting greater variability. The density of states, denoted by n ( E ), is a critical parameter in our study. We sum the density of states over all possible values of N i for i = 1, …, n to obtain the average density of states ⟨ n ( E )⟩ for the multi-state system, expressed as follows: where The coefficient C 0 ( N 0 ) represents the combinatorial expression for selecting N 0 random native residues from the total of N residues in the model protein system. The coefficient C i ( N i ) represents the combinatorial factor for selecting N i specific residues from the total N 0 native-like residues for each i = − 1, 2, …, n 1. The sum of all values of n ( E ) will receive significant contributions from various terms. However, in the thermodynamic limit, when N is large, the overall behavior of the model system is predominantly governed by the largest term in the expression in the logarithmic scale. In this context, log ⟨ n ( E )⟩ can be interpreted as the entropy for the microcanonical ensemble, as shown in Eq. 9 . Specifically, denotes the fraction of total native contacts for the entire chain, while represents the fraction of native contacts in the i th native state. Next, we derive the expressions for energy, entropy, and free energy per residue using the following thermodynamic relation, in conjunction with Eq.9 . The expression for energy is given by The substitution of the expression from Eq.10 into Eq.9 yields the entropy per residue. The resulting expression is given by: The per-residue Helmholtz free energy f is expressed as where ε and s are given by Eq.10 and Eq.11 from our model, respectively. Substituting these expressions into the equation leads to the following expression for f : The free energy functional f is minimized concerning ρ and ρ i separately to explore the phase diagram and understand the system’s free energy landscape [ 51 , 52 ]. The necessary partial derivatives are calculated as follows. From our model, we also get a critical temperature T 0 , where the system’s entropy becomes zero, entering a frozen state, which is also the second-order phase transition temperature for the system. Biasing a Single Native State over others In our model, we also examine the effect of stabilizing one of the native states compared to others by introducing a bias factor δ into the native energy term at the residue level for one distinct native state. This bias modifies the term for residue N 1 , thereby stabilizing one native state over others. By incorporating this bias, we can assess its effect on the partial derivatives of the free energy functional f with respect to ρ and ρ i , as shown in Equations 13 and 14. This analysis is crucial for understanding how stabilizing a single native state influences the order parameter and alters the phase diagram of our multi-native-state system. The modified relevant equations are derived and presented in S3 . Results & Discussion Expansion of unfolded region with increasing Native State Multiplicity The order parameters ρ and ρ i were obtained by solving Equations (13) and (14) simultaneously for systems with native states ranging from n = 1 to n = 5. In the case of n = 1, the model reduces to the single-state scenario (details provided in S1 ), while at n = 5, it represents a protein capable of adopting five distinct yet energetically equivalent native structures. We consistently used the parameter values of to solve the aforementioned equations. Additionally, we employed a threshold value of ρ = 0.75 beyond which the protein can be considered as folded, based on the criteria established by Wolynes and coworkers. [ 46 ]. The order parameter ρ exhibits two distinct ranges in all systems up to n = 5. Values of 0 < ρ ≤ 0.2 correspond to the unfolded state of the protein sequence, indicating low levels of native contacts. In contrast, values of 0.75 ≤ ρ ≤ 0.95 correspond to the folded state of our model sequence, reflecting the significant formation of native contacts. We constructed a phase diagram by plotting the control parameter on the x -axis and on the y -axis. The diagram reveals a clear phase transition line that separates the unfolded and folded regions, effectively distinguishing low and high values of the order parameter ρ across all scenarios( Fig. 1(a) ). Fig. 1(a) shows a gradual shift in the phase transition line and a contraction of the folded domain as the number of native states increases. The inset highlights x ∈ [2.5, 3.25] and y ∈ [0, 1.25], where the folded region enclosed by multistate systems is evident. As the complexity rises (i.e., more native states), the area of this folded region in the phase space diminishes, reflecting a lower propensity for the system to remain folded under otherwise identical thermodynamic conditions. In an additional plot where , we illustrate how ρ varies with the mean deviation of interaction energy, ( Fig. 1(b) ). This analysis identifies the critical value of at the onset of the folding transition. As illustrated in Fig. 1(b) , the onset points of the phase transition are 2.5, 2.65, 2.8, 2.95, and 3.05 for n = 1, 2, 3, 4, and 5, respectively, reflecting a progressive increase consistent with the contraction of the folded domain shown in Fig. 1(a) . Notably, at these onset points, the corresponding ρ values exceed 0.75, confirming that the overall sequence maintains an overall folded-like fraction of total native contacts. Download figure Open in new tab Figure 1. Exploration of protein folding transitions as system complexity increases. (a) Phase diagrams for one-to five-state systems, illustrating the contraction of the folded domain as the number of native states n increases. The inset highlights the region x ∈ [2.5, 3.25] and y ∈ [0, 1.25], showcasing the diminishing folded region in multistate systems. (b) The variation of the total native contact fraction ρ versus identifies the onset of the phase transition for each n . At these onset points, ρ for all systems surpass the dotted blue line representing the critical folding threshold ρ = 0.75. Folding Propensities of individual Native States in Multistate Systems Next, we determined the individual native contact values ρ i (with i = 1 to 5) while solving Equations (13) and (14), as detailed in the earlier section. Each ρ i represents the fractional native contacts associated with the i -th native state. In the single-state model, the overall fractional native contact ρ is identical to ρ 1 . However, in systems with multiple states, dividing both sides of Equation (4) by N yields the relationship For multi-state systems, the decomposition provides a systematic way to analyze how individual states contribute to the overall folding behavior of the system. We define the highest root as ρ 1 for all systems with varying n , followed by the second largest root as ρ 2 , and so on for subsequent roots ρ 3 , ρ 4 , etc. To evaluate the folding behavior of individual native states, we plot ρ 1 against the energy difference while keeping . This approach allows us to determine whether the highest ρ 1 value exceeds 0.75, serving as a criterion for identifying folded-like configurations of individual native states relative to the overall folding transition. Additionally, we analyze the variation of other roots against the same parameter to gain further insight into the folding scenario. Figure 2(a) presents ρ 1 as a function of for systems with one to five native states. The plot reveals that for n = 1 (single-state), n = 2 (two-state), and n = 3 (three-state) systems, ρ 1 surpasses the threshold of 0.75 at the transition points along the x-axis. This behavior signifies that at least one of the native states in these systems achieves a folded-like configuration. However, for n = 4 and n = 5 systems, where the sequence can adopt 4 or 5 equivalent native states, ρ 1 fails to cross the threshold, highlighting a notable change in the folded ensemble. While the overall native contact value ρ for all these systems confirms that the system predominantly resides in the folded region of the phase diagram(as shown in Figure 1(b) ), the behavior of ρ 1 reveals that for n = 4 and n = 5, none of the individual native states achieve a folded-like configuration. As shown in Figure S1 , the ρ vs. ρ 1 plot offers additional insight into multi-state folding. The downward trend in the ρ – ρ 1 curve for the four- and five-state systems indicates a partial “mixing” scenario, while up to three states exhibit single-state-like folding behavior. This highlights the more intricate folding landscape that emerges when multiple native states are accessible, as contributions to the overall native contacts are distributed among several possible configurations. Download figure Open in new tab Figure 2. Folding behavior of individual native states. (a) Variation of ρ 1 , the highest fractional native contact, as a function of . For n = 1, 2, and 3, ρ crosses the folding threshold indicated by the blue dotted line, while for n = 4 and n = 5, ρ 1 remains below this line. (b) Variation of ρ 2 , the second-largest fractional native contact, as a function of . An increasing trend in ρ is observed for n = 4 and n = 5, whereas for n = 2 and n = 3, ρ 2 remains relatively small, reflecting higher contributions from sub-dominant roots in systems with larger n . Figure 2(b) illustrates the second-largest root, ρ , as a function of for n = 2, 3, 4, and 5 native states. Upon moving from the unfolded to the folded regime, ρ 2 decreases to about 0.01 for n = 2 and n = 3, indicating a “de-mixing” scenario dominated by a single native state. In contrast, for n = 4 and n = 5, ρ 2 increases relative to its unfolded value, reaching approximately 0.07 and 0.08, respectively, consistent with a partial “mixing” where multiple states share contributions more uniformly. This trend underscores that, as the number of native states grows, no single conformation achieves a distinctly folded state, leading to a more evenly distributed set of native contacts. Folding Landscape for n = 2 System We derived the free energy, entropy, and internal energy expressions for systems with n native states, presented in Equations (12), (11), and (10), respectively. For the two native state systems ( n = 2), the expression of different terms with detailed derivations is provided in (S2) . In this case, the system is characterized by two roots, ρ and ρ 1 , related by ρ = ρ 1 + ρ 2 . The values of ρ and ρ 1 are determined by solving the coupled equations derived by setting n = 2 in Equations (13) and (14): By solving the coupled equations (17) and (18), we obtain a set of values for ρ and ρ 1 . The folding propensity for the two native-state system, in terms of ρ and ρ 1 , has been discussed in detail in Figures 1(b) and 2(a), respectively, against the relevant control parameters. For the two-state system, the onset point is found to be 2.65 with , marking the start of the folding transition. We investigate the folding behavior of a two-state system by examining the free energy functional f , plotted on the z -axis, as a function of the fractional native contacts ρ (on the x -axis) and ρ 1 (on the y -axis), which together characterize the system’s transition between the unfolded and folded states. The analysis is conducted at two distinct values of the control parameter and 2.9, while keeping . These parameter values are strategically chosen to capture the behavior on either side of the transition point : 2.4 in the unfolded regime and 2.9 in the folded regime. Figure 3 showcases the free energy surface of the two native-state systems, capturing its evolution across the unfolded, transition, and folded regimes. Figure 3(a) represents the unfolded regime, where the free energy surface reveals a single stable unfolded state coexisting with two equally stable folded states. Figure 3(b) illustrates the free energy surface at the transition point, where the unfolded and folded states are energetically equivalent. Figure 3(c) corresponds to the folded regime, where the two folded states are energetically more stable than the unfolded state. The free energy surface clearly shows two distinct folded basins: one at ( ρ, ρ 1 ) ≈ (0.868, 0.856) and another at ( ρ, ρ 1 ) ≈ (0.868, 0.012). It follows that for ρ 1 = 0.012, ρ 2 = 0.856. Therefore, the two folded basin are symmetric to each other in terms of order parameters value. For systems with n native states, the folded basin will have n minima which are symmetric with respect to order parameter values. Download figure Open in new tab Figure 3. Free energy surfaces depicting the evolution from unfolded to folded regimes. (a) A dominant unfolded basin; (b) the emergence of two folded basins at the transition point; (c) The folded regime is characterized by two equally stable folded basins. Modification of the Entropically Frozen Region of the Phase Diagram In our phase diagram, the second-order phase transition line emerges when the order parameters ρ, ρ i , and Δ L/T satisfy the conditions of Equation (11) for an n -dimensional system, resulting in zero entropy per residue. Under these conditions, the system enters an entropically frozen state. To illustrate this behavior, we compare two representative systems: one with a single native folded state and another with two coexisting native folded states. The transition temperature for the one-state system is given by (Eq.(4) in S1) , while that for the two-state system is provided by (Eq.(8) in S2) . We systematically compare first-order and second-order phase transitions within a two-dimensional parameter space defined by and . Figure 4 illustrates distinct phase-transition behaviors for systems with one and two native folded states. For n = 2, both the folded and frozen regions contract: the red and black lines depict first-order phase boundaries (with the black line corresponding to the two native state system), and its rightward shift indicates contraction of the folded region. The orange boundary, represent-ing the second-order transition line for n = 2, shifts leftward—indicating contraction of the frozen region—while the green boundary corresponds to the n = 1 system. A shift toward a higher value of implies a lower transition temperature, and vice versa. Thus, our results indicate a gradual lowering of the folding temperature and an increase in the frozen transition temperature with increasing n , consistent with the notion that larger n entails greater frustration. Download figure Open in new tab Figure 4. Comparative phase transition analysis for first- and second-order transitions. Biasing one native state in n=4 system We are interested in the four-state system ( n = 4), where the transitions of ρ and ρ 1 have been previously examined in Figures 1(a) and 2(b) . It was observed that while ρ undergoes a transition into the folded regime at the onset of fails to cross the folding threshold of 0.75, indicating that none of the individual native states achieve a folded-like configuration. To evaluate the impact of biasing one native state out of the possible four, we introduce an additional stability factor δ to the residue level interaction energy term ε − ε 0 of one native state. This controlled bias alters the stability of the selected native state relative to others, as reflected in the modified free energy functional ( Eq.(9) in S3 ). We determine the modified roots, ρ and ρ i , by solving the three coupled equations ( Equations (10), (11), and (12) in S3 ),with the parameter δ explicitly included in the corresponding equation. This approach allows us to assess how the perturbation affects the stability and native contact values of individual states. Figure 5(a) displays the phase diagram for both the biased and unbiased ( n = 4) systems at δ ≈ 0.4. The perturbation shifts the phase transition line to the left, resulting in an expanded folding region compared to unbiased one. The larger folding region in the perturbed system suggests an increased folding propensity. Additionally, Figure 5(b) shows that the perturbed ρ 1 exceeds the critical threshold of 0.75, while the unperturbed ρ 1 remains below this value. This indicates that perturbations enhance the folding propensity of selected individual native state and folds into that particular native state. Download figure Open in new tab Figure 5. (a) Phase diagrams for four-state systems under biased and unbiased conditions. The main panel shows that bias shifts the phase transition line leftward and enlarges the folding region. The inset compares the biased state (black line with blue folded region) to the unbiased state (red line with green folded region), with the blue region being notably larger. (b) ρ 1 transitions show that only the perturbed system’s ρ 1 crosses the blue dotted folding cutoff line, indicating enhanced folding propensity. Discussion and Conclusion Our study demonstrates that increasing the number of native folded states in protein models leads to a contraction of the folded domain, indicating a decrease in the folding propensity of the system. Additionally, there are two scenarios that exist. In the first scenario, for n = 1 to n = 3, where both overall native fraction ρ and the highest individual native fraction, ρ 1 , surpass the folding threshold of 0.75. More specifically, there exist two folded minima in the n = 2 system and three folded minima in the n = 3 system where a fraction of native contacts of one distinct native state crosses the threshold value. This scenario can be related to the proteins having more than one folded state as in the metamorphic proteins. In the second scenario, for n = 4 and n = 5, the expansion of the unfolded region becomes quite significant. This suggests that such sequences with larger values of n will find themselves in the unfolded region of the phase diagram under physiological thermodynamic conditions. This scenario parallels with the intrinsically disorder proteins where the configurations remain unfolded-like in physiological conditions. Additionally, in the folded configuration of such sequences, none of the individual native states achieve a distinctly folded configuration. To examine the effect of selective stabilization, we introduce a targeted stabilization of one particular native state in the n = 4 system. This modification, results in expansion of the folded region in the phase diagram compared to the unperturbed case, increasing the folding propensity of the sequence. Additionally, the fractional native contact value of the stabilized native state exceeds the folding threshold, a scenario absent in the unperturbed system, indicating a folded configuration similar to n = 1 case. This result demonstrates how selective stabilization can enhance folding propensity at the level of individual native states, offering a potential understanding of the folding upon binding scenario of the intrinsically disorder proteins. Our study provides a theoretical framework for understanding folding scenarios in protein system with multiple possible native states, emphasizing how native-state multiplicity shapes folding phase diagram, metamorphic scenario and the coupled folding upon binding scenario of the IDPs. Conflicts of Interest The authors declare that they have no conflict of interest with respect to the publication of this article. 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