Is Catalytic Activity of Chaperones a Selectable Trait for the Emergence of Heat Shock Response?

Abstract

Although heat shock response is ubiquitous in bacterial cells, the underlying physical chemistry behind heat shock response remains poorly understood. To study the response of cell populations to heat shock we employ a physics-based ab initio model of living cells where protein biophysics (i.e., folding and protein-protein interactions in crowded cellular environments) and important aspects of proteins homeostasis are coupled with realistic population dynamics simulations. By postulating a genotype-phenotype relationship we define a cell division rate in terms of functional concentrations of proteins and protein complexes, whose Boltzmann stabilities of folding and strengths of their functional interactions are exactly evaluated from their sequence information. We compare and contrast evolutionary dynamics for two models of chaperon action. In the active model, foldase chaperones function as nonequilibrium machines to accelerate the rate of protein folding. In the passive model, holdase chaperones form reversible complexes with proteins in their misfolded conformations to maintain their solubility. We find that only cells expressing foldase chaperones are capable of genuine heat shock response to the increase in the amount of unfolded proteins at elevated temperatures. In response to heat shock, cells’ limited resources are redistributed differently for active and passive models. For the active model, foldase chaperones are overexpressed at the expense of downregulation of high abundance proteins, whereas for the passive model; cells react to heat shock by downregulating their high abundance proteins, as their low abundance proteins are upregulated.

Introduction

From transcription to folding, errors emerging from almost all steps of protein synthesis render protein translation energetically most costly cellular process (1,2). Furthermore, protein misfolding decreases the ability of folded proteins to function and engage in functional protein-protein interactions (PPIs) and increases the probability of formation of nonfunctional protein-protein interactions (NF-PPIs) and even of toxic insoluble aggregates (3–7). Hence, high-fitness costs of erroneous proteins synthesis and strong selective pressure against protein misfolding severely constrain protein evolution (8,9).

Chaperones, also known as heat-shock proteins, play a crucial role in maintaining protein homeostatic balance by replenishing the pool of folded proteins at desired levels and thus preventing protein aggregation as well as sequestration of functional proteins in NF-PPIs (10,11). Chaperones are also endowed with other important functions such as assembly of protein complexes (12), stabilization of PPIs (12), and protein translocation in cytoplasm (13). Importantly, chaperones have been shown to function as phenotypic capacitors (14), which buffer destabilizing effects of deleterious mutations (15), promote genetic diversity, and thus speed up adaptive evolution (16–18). Genome-wide studies also showed that chaperone clients exhibit higher substitution rates than nonclients, indicating an innovative potential in the chaperone-mediated evolution of proteins (19,20).

Recently, we developed a physics-based ab initio multiscale evolutionary model of living cells to investigate the role of chaperones in adaptive protein evolution via population dynamics simulations (22). Our cell model provided a unified theoretical framework that helped to understand and explain a number of important effects regarding the chaperone-mediated protein evolution. In accord with in vivo evidence obtained in our lab (23), our theoretical results showed that the chaperon action in cytoplasm is highly pleiotropic: by rescuing proteins trapped in unfolded conformations, chaperons dramatically decrease their sequestration in NF-PPIs and thus mediate their accessibility to proteases (23). Moreover, our model predicted in an experimentally testable way how chaperones buffer deleterious effects of mutations and accelerate adaptive molecular evolution by amplifying multiscale epistatic effects among consecutive mutations (22).

In this study, we extend our previous analysis of chaperone-mediated protein evolution to study the emergence of heat shock response observed in bacterial cell populations (24,25). Following our previous work (22), we consider two different mechanisms of chaperone action: an active mechanism to model the foldase chaperones and a passive mechanism to model the holdase chaperones. The active chaperones, i.e., foldases in our model affect the ratio between folding and unfolding rates and shift the equilibrium toward folded protein species. The foldase chaperones such as Hsp60—i.e., GroEL in Escherichia coli (E. coli)—effectively utilize the active mechanism and function as out-of-equilibrium molecular folding machines (26,27). However, some other chaperones such as hdeA/B or Hsp33 in E. coli presumably act passively as holdases to prevent an irreversible aggregation by reversibly binding to unfolded proteins (28,29).

Our previous study showed that only the foldase chaperones can accelerate the rate of protein evolution in adapting cell populations (22). In line with this result, in this study we only observe a genuine heat shock response for cells equipped with the foldase chaperones. Our results also indicate that the heat shock response orchestrated by foldase chaperones is in fact an unfolded protein response that manifests in a temperature dependent manner (22). For both models, irrespective of chaperone action, cells react to heat shock by redistributing cells’ limited resources. For the active model, upon heat shock, cells reduce the concentration of their most abundant protein at the expense of increasing the concentration of their foldase chaperones. In the passive model, the holdase chaperones do not play a crucial role; nevertheless, to cope with heat shock, cells again decrease the concentration of their most abundant protein to increase the concentrations of their less abundance proteins, which function by forming dimers. Hence, our results demonstrate, to our knowledge, two unique evolutionary cost-benefit trade-offs, one originating directly from a catalytic activity of chaperones, and the other emerging when chaperone machinery plays no role in restoring protein homeostatic balance under the heat shock stress.

Materials and Methods

Active and passive models of chaperone action

We assume that each protein in our cell model folds with a simplified two-state folding kinetics:

$$U_i\underset{k_u^i}{\overset{k_f^i}{\rightleftarrows }}{F}_i.$$ (1)

We incorporate the chaperone action into our cell model by coupling the chaperone activity with two-state protein folding kinetics by using two different chaperone models. In the active model, the foldase chaperones function as a catalyst that effectively accelerates the rate of protein folding without changing the rate of unfolding:

$$U_i+Ch\stackrel{K_{Ch}^i}{\rightleftarrows }[U_i\cdot Ch]\stackrel{k_{Ch}^i}{\to }{F}_i+Ch.$$ (2)

In the passive model, the holdase chaperones interact with proteins in their unfolded/misfolded states to form soluble complexes without affecting the actual rate of protein folding:

$$U_i+Ch\stackrel{K_{Ch}^i}{\rightleftarrows }[U_i\cdot Ch]$$ (3)

We modeled the interactions of chaperones with unfolded proteins by a 3×3 two-dimensional (2D) square lattice that includes nine amino acid residues (YLVAFAVYF), which are essential for substrate binding and found in the apical domain of chaperonin GroEL (30).

As it is well known, by reducing the activation barrier, a conventional catalyst accelerates both forward and backward reactions. In this respect, the foldase chaperones can be considered as one-way catalysts—out-of-equilibrium folding machines that consume ATP to accelerate only the forward reaction leading to the increase in the amount of folded proteins without affecting the rate of backward reaction by which proteins unfold. In our chaperone models, we used a preequilibrium assumption to model interactions of proteins in their unfolded states with chaperones. This is because binding and release of unfolded proteins to chaperones occur very fast as compared with the rest of rate limiting kinetics steps, involved in chaperone-assisted protein folding, one of which almost always requires ATP hydrolysis. As further noted in the discussions, in our model, molecular binding, folding, and unfolding events are all coarse-grained. Therefore, the biologically relevant net effect of foldase action in our model is the effective increase in concentrations of folded proteins (22). Our active model directly applies to the foldase chaperones such as the chaperonin GroEL in prokaryotes, the catalytic activity of which is well established (26,27,31).

Intra- and intermolecular interactions

Throughout all our simulations we followed the methods reported in our previous work (22), which we briefly summarize here. Our ab initio cell model has a six-loci explicit genome that translates six essential 27-mer 3×3×3 cubic lattice proteins (21,32). We model inter- and intramolecular interactions made by the lattice proteins with M.J. potentials (33). The 27-mer lattice model has a complete set of 103,346 maximally compact conformations (32). For our computational ensemble, however, we only use a random subset of 10,000 conformations. For each protein $i=1,\mathrm{\dots },6$ we calculate the Boltzmann probability of folding to native state, i.e., $P_{nat}^i$ by the following,

$$P_{nat}^i=\frac{\exp [-E_1^i/T]}{{\sum }_{k=1}^{10000}\exp [-E_k^i/T]},$$ (4)

where $E_1^i$ is the energy of ground state, i.e., the native state, $E_k^i$ are the energies of higher states, and T is the temperature in units calibrated to M.J. potentials. In our cell model, folded proteins and proteins in unfolded states interact with each other to form functional PPIs and NF-PPIs. Also, both types of chaperones form complexes with unfolded proteins. We used a rigid docking procedure to model all these molecular interactions. There are 144 docking modes for a dimer molecule. However, for a chaperone-unfolded protein complex, there are only 24 binding modes since we model chaperones with a 3×3 2D lattice surface. By using these binding modes, for example, we calculate the Boltzmann probability of interaction $P_{\mathrm{int}}^{ij}$ between two dimeric proteins i and j by using the following formula,

$$P_{\mathrm{int}}^{ij}=\frac{\exp [-E_f^{ij}/T]}{{\sum }_{k=1}^{144}\exp [-E_k^{ij}/T]},$$ (5)

where $E_k^{ij}$ are the interaction energies of each binding mode for $k=1,\mathrm{\dots },144$, $E_f^{ij}$ is the interaction energy for the preset functional binding mode, and T is the temperature. Similarly, we calculate the binding constant $Q_{U_i^{n}Ch}$ for a protein “i” in unfolded state “n,” in complex with the chaperone Ch as follows,

$$Q_{U_i^{n}Ch}={\sum }_{k=1}^{24}\exp [-E_{U_i^{n}Ch}^k/T],$$ (6)

where $E_{U_i^{n}Ch}^k$ are the binding energies for the complex; see (22) for details.

Simulation procedure

By using the above procedure for interaction energies, the equilibrium constants for any binary protein complex formed by folded-folded, folded-unfolded, unfolded-unfolded, and unfolded-chaperone reactions can be calculated (22). Recently, we developed a mean-field approximation to calculate equilibrium constants for such reactions (22). Generally, the law of mass of action (LMA) coupled differential equations are hard to solve by direct integration schemes because these equations are inherently nonlinear. However, very accurate solutions can still be obtained by iterative methods such as the one used in previous studies (21,34). We developed an iterative algorithm that can accurately solve not only LMA equations involving equilibrium reactions between different molecular species, but also LMA equations involving additional reactions between conformational isomers of the same molecular species (22). This new algorithm is a straightforward generalization of the existing iterative algorithms and used throughout in this work to solve the LMA equations. In all our simulations we used a population size of 1,000 cells. To generate stochastic evolutionary trajectories we employed a variant of Gillespie algorithm reported in detail in a previous study (22). The algorithm maintains the constant population size by replacing a newborn cell with a randomly chosen cell in population. The cell division occurs via semi-conservative replication such that a mother cell gives birth to a daughter cell and one of the following events can happen: either a mutational event with constant probability m = 0.001 per gene per replication or an event that changes the expression level of one of randomly chosen proteins or chaperone with a constant rate e.r. = 0.01 per cell division such that the total concentration of protein “i” in a new born cell is derived from that of an old cell by the formula $\left[C_i^{new}\right]=\left[C_i^{old}\right](1+\epsilon )$ where $\epsilon$ is a Gaussian random number with zero mean and variance of 0.1.

Results

Our ab initio cell model consists of an explicit six-loci genome that encodes six 27-mer-lattice proteins (32), each of which is essential, and thus controls the cell division rate. A number of previous studies (21,22,34,35) demonstrated that this coarse-grained model provides computational advantages in calculating folding and binding energies of proteins exactly for a selected representative subset of conformations. In this study, following an earlier work (22), we employ several well-justified assumptions to make our simulations computationally tractable. We define the minimum energy state of proteins as native state and assume that proteins need to fold to a native state to function. At the beginning of our simulations we assign the functional native conformations of our proteins and functional predetermined docking modes of protein complexes (see Materials and Methods). We take into account the unfolded or misfolded states of proteins via a mean field approach (22) by treating the ensemble of unfolded conformations as a set of maximally compact yet nonnative conformations.

Proteins in all states, i.e., folded or unfolded, can interact with each other in the cytoplasm of model cells to engage in functional as well as nonfunctional PPIs, see Fig. 1. However, we model the chaperone interaction network by using only unfolded proteins, as shown in Fig. 1 B. This is because it has been found that most chaperones rarely interact with proteins in their native states (36–38). To define the fitness function for our cell model, we postulate a functional PPI network, consisting of prototypical examples of commonly known network motives, see Fig. 1 A. We assume that the first protein is active as monomer, the second and third proteins form a heterodimer to function, and finally, the fourth, fifth, and sixth proteins form a date triangle consisting of three functional heterodimers. Hence, for the postulated PPI network, the fitness function that determines the cell division rate is as follows:

$$b=b_0\frac{G_1{G}_{23}{\left(G_{45}{G}_{46}{G}_{56}\right)}^{1/3}}{1+\alpha {\left({\sum }_{i=1}^7{C}_i-C_T\right)}^2},$$ (7)

where $b_0$ is used as a parameter to scale the rate and time. The functional concentrations of monomer and dimers in Eq. 7 are defined as $G_1=\left[F_1\right]\text{and}{G}_{ij}=[F_i\cdot F_j]\times P_{\mathrm{int}}^{ij}$, respectively, where $\left[F_1\right]$ is the functional concentration of monomeric protein in its native conformation, $[F_i\cdot F_j]$ is the functional concentration of heterodimer formed by proteins i and j in their native forms and $P_{\mathrm{int}}^{ij}$ is the Boltzmann probability that the proteins i and j form a functional complex in a predefined correct docking conformation (see Materials and Methods). In Eq. 7, $C_T$ is the sum of initial concentrations of proteins and $C_i$ is the total individual concentrations of proteins in time for $i=1,\mathrm{\dots },7$, including the chaperone, $C_{Ch}=C_7$. Overall, the role of denominator in Eq. 7 is to penalize the deviations from the optimum protein levels to avoid a fitness gain by a mere overexpression of proteins. Hence, we determine the cell division rate by a bottleneck-like fitness function, which stems from the intuitive physical-biological assumption, supported by the experiments (23), that the fitness (i.e., division rate) of a cell depends on the functional concentrations of its proteins and proteins complexes.

Figure 1.

An illustration of molecular interactions in model cells and protein folding kinetics coupled with chaperone actions. (A) A functional PPI network showing only folded proteins (blue cubes) and their interactions (blue lines). (B) Interaction network of chaperone (green square) includes only unfolded proteins. (C) Passive model where $K_{Ch}^i$ is the equilibrium constant for the complex formed between holdase chaperone and the unfolded protein “i.” (D) Active model for foldase chaperones, where $k_{Ch}^i$ is the rate constant for forward reaction, where actual protein folding occurs for the unfolded protein “i.” To see this figure in color, go online.

We carried out evolutionary simulations to investigate the emergence of heat shock response in cell populations. Each set of our simulation consists of 100 independent evolutionary trajectories. We present our results as mean values averaged over these 100 evolutionary trajectories. We start our simulations from the same monoclonal population, whose protein sequences were designed to be stable, i.e., $P_{nat}>0.8$ in its respective native conformation, where $P_{nat}$ stands for the Boltzmann probability of folding to a native state. We employed the design procedure (39) to generate these stable initial protein sequences for our evolutionary simulations. No other constraints, including solubility were imposed on the initial sequence design. Note also that our choice of initial conditions for protein stabilities, i.e., starting the simulations from initial stable sequences is just a convention that we used to reduce the initial equilibration time and does not serve for any another purpose. That is, starting simulations from less-stable protein sequences $P_{nat}<0.8$ would be as good as starting from stable sequences $P_{nat}>0.8$ for our purpose because we allow proteins sequences to adapt for sufficiently long time.

Initially, at low temperature, i.e., T = 0.85 (in units calibrated to Miyazawa-Jernigan potentials (33)), we propagated our cell populations for 200,000 generations until cells acquired considerable fitness gain and steady-state mutational load. We then simulated the heat-shock response by taking into account three different temperatures jumps (i.e., up- and downshifts). More specifically, at the end of initial evolutionary equilibration period of 200,000 generations, we exposed the evolved populations instantaneously to an elevated temperature of 1) $T=0.85\to 0.95$, 2) $T=0.85\to 1.05$, and 3) $T=0.85\to 1.15$ for 5000 generations; and then we dropped the temperature instantaneously back to its original value of 1) $T=0.95\to 0.85$, 2) $T=1.05\to 0.85$, and 3) $T=1.15\to 0.85$ and let these populations evolve for 10,000 generations further at the original low temperature.

In the following, we report the time evolution of a number of quantities such as cellular fitness (cell division rate), the total protein as well as chaperone concentrations, and the fraction of protein material wasted in NF-PPIs. It is noteworthy that all these quantities are the double-mean values, i.e., the population means (averaged over cell populations) are averaged again over 100 independent evolutionary trajectories.

Initial equilibration period: foldase vs. holdase chaperones

For the initial equilibration period of 200,000 generations, we plot the time evolution of birth rates b and total chaperone concentrations $C_{Ch}$ in Fig. 2, A and B, respectively, for both chaperone models. As seen in Fig. 2 A, at the beginning of our simulations, the fitness values are extremely small for both chaperone models. This is because we start our simulation from stable protein sequences, the interaction surfaces of which are not optimized to strengthen functional PPIs, which constitute the crucial components of the fitness function, as given in Eq. 7, and therefore, the initial concentrations of functional dimers are very small and so are the rates at which cells divide.

Figure 2.

The time evolution of the mean fitness b and mean chaperone concentrations $C_{Ch}$ for T = 0.85 in the initial stage of adaptation for 200,000 generations. (A) The time evolution of b for the active model, i.e., foldase chaperones (red lines), and the passive model, i.e., holdase chaperones (blue lines). (B) The time evolution of total concentration of chaperone $C_{Ch}$ for the active model, i.e., foldases (red lines), and the passive model, i.e., holdases (blue lines). To see this figure in color, go online.

We see in Fig. 2 B that the total chaperone concentrations drop very fast initially within 20,000 generations for both the foldase and holdase chaperones. Our simulations started from equal protein and chaperone concentrations, $C_i(t=0)=C_{Ch}(t=0)=0.1$. Within the initial 20,000 generations, the total protein concentrations including chaperones are evolutionarily optimized; that is, cells’ resources are redistributed to maximize the cellular fitness. Apparently, at the initial stage of evolutionary dynamics, it is more beneficial for cells with very low fitness to directly harness the cells’ limited resources to increase the amount of functional proteins (i.e., products of essential genes one to six) rather than increasing chaperon expression since maintaining a high chaperone concentration does not initially increase appreciably the abundance of folded proteins. However, as organisms evolve to greater fitness values, it becomes more and more important to maintain a certain level of foldase chaperones as a buffer since the foldase chaperones speed up the rate of evolution in adapting cell populations whereas the holdase chaperones do not, as shown in our previous study (22). Indeed, at the end of 200,000 generations time evolution, we see that the fitness of cells containing foldase chaperones is almost three times greater than that seen for the model cells with holdase chaperones, indicating the evolutionary benefit provided by the foldase chaperones.

Emergence of the heat shock response

In the next figures, we present our results by focusing on the evolutionary dynamics during and after the heat shock. We plot the time evolution of average birth rate b in Fig. 3, A and B, and the time evolution of total chaperone concentrations $C_{Ch}$ in Fig. 3, C and D for the foldase and holdase chaperone models, respectively. The fitness plots display an instantaneous drop of fitness following the temperature upshift for both chaperone models. We discuss in detail in the next section that the rise in temperature results in an abrupt increase in the amount of unfolded proteins, which in turn leads to the observed instantaneous drop of fitness. As seen in Fig. 3, A and C, the cells with foldase chaperones respond to the heat shock very rapidly by increasing the concentration of their foldase chaperones. In what follows, the rapid upregulation of foldase chaperones leads to a swift recovery of cellular fitness in a temperature dependent manner. In contrast, as seen in Fig. 3, C and D, we observe neither any recovery of cellular fitness nor any increase in the levels of holdase chaperones in the cells modeled with passive mechanism. Note that the scale of y axis is very different in Fig. 3, C and D for foldase and holdase concentrations. Also, although $C_{Ch}$ for foldase shows a steady-state behavior, $C_{Ch}$ for holdase seemingly does not. We used such a small scale for the holdase chaperones to emphasize the actual order of magnitude of $C_{Ch}$. The fluctuations in $C_{Ch}$ seen in Fig. 3 D shows that the steady state is in fact a dynamic equilibrium, and $C_{Ch}$ fluctuates with a Gaussian white noise modulating $C_i$ for i = 1…6 and including $C_{Ch}$ (see Materials and Methods), which is highly noticeable at such small scale.

Figure 3.

The time evolution of the mean fitness b and mean chaperone concentrations $C_{Ch}$ for T = 0.85 (black lines), T = 0.95 (red lines), T = 1.05 (green lines), and T = 1.15 (blue lines). (A) The time evolution of b for the active model, i.e., foldase chaperones. (B) The time evolution of b for the passive model, i.e., holdase chaperones. (C) The time evolution of $C_{Ch}$ for foldase chaperones. (D) The time evolution of $C_{Ch}$ for holdase chaperones. To see this figure in color, go online.

At the end of heat shock stress, i.e., right after the temperature drops back to the original value T = 0.85 at 205,000 generations, the fitness stays low and does not reach its original value, just before the heat shock at 200,000 generations. This is because the cells’ resources (i.e., total concentrations of gene products) at this junction are optimized for the elevated temperatures of heat shock but not for the original low temperature. As we show in the next sections, however, the cells optimize the use of their resources (i.e., protein materials) by adjusting the concentrations of each protein before and after the heat shock to maximize the cellular fitness.

Hence, our model predicts that the catalytic activity of foldase chaperones, which renders chaperone upregulation upon heat shock stress beneficial, is a selectable trait for short-time adaption of cells to elevated temperatures, i.e., for the emergence of heat shock response.

Heat shock is an unfolded protein response

To determine the sequence of elementary events that cause upregulation of foldase chaperones upon the heat shock, we analyze the time evolution of a number of molecular quantities that involve unfolded proteins species. Since we only consider binary PPIs, an unfolded protein in the cytoplasm of our model cells can be found as a free monomer $\left[U_i^{free}\right]$, in complex with chaperones $\left[U_{i}Ch\right]$, in complex with other unfolded proteins $\left[U_i{U}_j\right]$, and in complex with proteins in their native form $\left[U_i{F}_j\right]$ where indices run as i,j = 1…6. The total concentration of unfolded proteins can then be expressed as the sum of all these unfolded proteins species, i.e., ${\left[U\right]}_{tot}={\left[U^{free}\right]}_{tot}+{\left[UCh\right]}_{tot}+{\left[UU\right]}_{tot}+{\left[UF\right]}_{tot}$ where ${\left[U^{free}\right]}_{tot}={\sum }_{i=1}^6\left[U_i^{free}\right]$ is the total concentration of free unfolded proteins (i.e., unfolded proteins in monomeric form), ${\left[UCh\right]}_{tot}={\sum }_{i=1}^6\left[U_{i}Ch\right]$, is the total concentration of chaperone-unfolded protein complexes, and ${\left[UU\right]}_{tot}={\sum }_{i=1}^6{\sum }_{j\ge i}^6\left[U_i{U}_j\right]$ is the total concentration of protein complexes between two unfolded proteins, and finally ${\left[UF\right]}_{tot}={\sum }_{i=1}^6{\sum }_{j\ge i}^6\left[U_i{F}_j\right]$ is the total concentration of protein complex formed by unfolded and folded proteins.

We present the time evolution of ${\left[U\right]}_{tot}$ for the foldase chaperones in Fig. 4 A, and for the holdase chaperones in Fig. 4 B, respectively. The time evolution of the components of ${\left[U\right]}_{tot}$ is given in Figs. S1 and S2 in the Supporting Material for the active and model models, respectively. Following the temperature upshift, Fig. 4, A and B show an instantaneous increase in the concentration of unfolded proteins for both chaperone models. This combined effect is because of the rapid increases in the concentrations of free unfolded proteins ${\left[U^{free}\right]}_{tot}$ as well as unfolded protein complexes of the form ${\left[UU\right]}_{tot}$ and ${\left[UF\right]}_{tot}$, as seen in Figs. S2 and S3 for both models. Our results indicate that only cells with foldase chaperones are capable of responding to the increase in ${\left[U\right]}_{tot}$ by rapidly upregulating the concentration of their active chaperones to rescue the cellular fitness. Because of heat shock response, as the foldase chaperone concentration increases, they start forming complexes with unfolded proteins, as can be seen from the immediate increase in the concentration of ${\left[UCh\right]}_{tot}$ and we do also observe a rapid decrease in ${\left[U^{free}\right]}_{tot}$ and fast decline of ${\left[UU\right]}_{tot}$ and ${\left[UF\right]}_{tot}$ (see Fig. S1). For the passive model, we also observe a rather small decline of ${\left[U^{free}\right]}_{tot}$ and ${\left[UU\right]}_{tot}$ during the evolution at elevated temperatures. Nevertheless, this effect is not related to heat shock response of foldase chaperones since neither $C_{Ch}$ nor ${\left[UCh\right]}_{tot}$ change appreciably for the passive model, when the cells exposed to the elevated temperatures. As we shall see in the next section, this effect in fact originates from the redistribution of cells resources rather than the chaperones’ response to the increase in ${\left[U\right]}_{tot}$.

Figure 4.

The time evolution of the mean value of total unfolded protein concentrations for T = 0.85 (black lines), T = 0.95 (red lines), T = 1.05 (green lines), and T = 1.15 (blue lines). The time evolution of the mean value of the total concentration of unfolded proteins ${\left[U\right]}_{tot}={\left[U^{free}\right]}_{tot}+{\left[UCh\right]}_{tot}+{\left[UU\right]}_{tot}+{\left[UF\right]}_{tot}$ is given in (A) for the foldase chaperones and in (B) for the holdase chaperones. See the main text for the definitions of ${\left[U\right]}_{tot},{\left[UCh\right]}_{tot},{\left[UU\right]}_{tot}$ and ${\left[UF\right]}_{tot}$. To see this figure in color, go online.

Since the heat shock stress is a temporary exposure of cells to elevated temperatures, the cells’ response to heat shock should be regulated through a reversible mechanism. Indeed, our results show that at the end of heat shock stress, when the temperature is switched back to its original value, the concentrations of all unfolded molecular species return to their original values right before the heat shock, and for the foldase model, the chaperone concentration decreases so that this extra resource gained by the downregulation of chaperones can be used for other essential proteins to increase the fitness.

The effect of temperature increase on cellular fitness could be severe even though its effect on the amount of unfolded proteins might not appear too dramatic. This is because protein misfolding in a cell creates a domino effect in which seemingly small amount of unfolded proteins suffice to generate vast increases in NF-PPIs, which in turn could trap an enormous amount of protein materials. Therefore, to observe the effectiveness of chaperones upon heat shock, it is also informative to follow in tandem the time evolution of the fraction of total amount of unfolded proteins in the cell and the fraction of proteins sequestered in the form of NF-PPIs.

Keeping this in mind, in Fig. 5, we analyze the time evolution of the fraction of proteins sequestered in the form of NF-PPIs for both the foldase and holdase chaperones. Specifically, we plot the time evolution of ${\mathrm{\Gamma }}_m={\mathrm{\Phi }}_1/C_1$ for the monomers in Fig. 5, A and D, and ${\mathrm{\Gamma }}_h=({\mathrm{\Phi }}_2+{\mathrm{\Phi }}_3)/(C_2+C_3)$ for the heterodimers in Fig. 5, B and E, and ${\mathrm{\Gamma }}_d=({\mathrm{\Phi }}_4+{\mathrm{\Phi }}_5+{\mathrm{\Phi }}_6)/(C_4+C_5+C_6)$ for the date triangle proteins in Fig. 5, C and F, for the foldase and holdase chaperones, respectively. Note that $C_i$ stands for the total concentration of protein i and ${\mathrm{\Phi }}_i$ for the total concentration of protein i engaged in NF-PPIs. The same quantities for the initial 200,000 generations are given in Fig. S3.

Figure 5.

The time evolution of the mean value of the fractions of proteins involved in NFP-PPIs to their total concentration for T = 0.85 (black lines), T = 0.95 (red lines), T = 1.05 (green lines), and T = 1.15 (blue lines). In (A), (B), and (C), we present our results for the active model, i.e., foldase chaperones, and in (D), (E), and (F) for the passive model, i.e., holdase chaperones. (A) and (D) show the time evolution of ${\mathrm{\Gamma }}_m={\mathrm{\Phi }}_1/C_1$ for the functional monomer. (B) and (E) show the time evolution of the average NF-PPI ${\mathrm{\Gamma }}_h=({\mathrm{\Phi }}_2+{\mathrm{\Phi }}_3)/(C_2+C_3)$ for heterodimers. (C) and (F) show the time evolution of the average NF-PPI for ${\mathrm{\Gamma }}_d=({\mathrm{\Phi }}_4+{\mathrm{\Phi }}_5+{\mathrm{\Phi }}_6)/(C_4+C_5+C_6)$ date triangles, where

$$\begin{array}{l}{\mathrm{\Phi }}_1=C_1-\left[F_1\right]-\left[U_1\right]-[U_1\cdot Ch],{\mathrm{\Phi }}_2=C_2-G_{23}-\left[F_2\right]-\left[U_2\right]-[U_2\cdot Ch],\hfill \\ {\mathrm{\Phi }}_3=C_3-G_{23}-\left[F_3\right]-\left[U_3\right]-[U_3\cdot Ch],{\mathrm{\Phi }}_4=C_4-G_{45}-G_{46}-\left[F_4\right]-\left[U_4\right]-[U_4\cdot Ch],\hfill \\ {\mathrm{\Phi }}_5=C_5-G_{45}-G_{56}-\left[F_5\right]-\left[U_5\right]-[U_5\cdot Ch],{\mathrm{\Phi }}_6=C_6-G_{46}-G_{56}-\left[F_6\right]-\left[U_6\right]-[U_6\cdot Ch].\hfill \end{array}$$

To see this figure in color, go online.

At the beginning of the initial 200,000 generations time evolution, most proteins are wasted to NF-PPIs (Fig. S3) because we only optimized our protein sequences for the stability and not the strength of their functional PPIs. However, the coevolution with foldase chaperones caused a rapid recovery of proteins engaged in NF-PPIs, which in turn increased the cellular fitness and resulted in a faster evolution of cells with active model than that of passive model. As seen in Fig. 5, A–C, following the temperature upshift, clearly, the rapid increase in the concentration of foldase chaperones in response to the heat shock rescues cellular fitness very fast by decreasing the amount of misfolded proteins. This effect is highly pronounced for heterodimers and date triangle proteins as compared with the monomer. This is because the monomeric protein has already evolved to high stabilities and is evolutionarily optimized to avoid NF-PPIs. On the other hand, the heterodimers and date triangle proteins have still been evolving to reach their phenotypic target that optimally contributes to cellular fitness and therefore these proteins are more prone to NF-PPIs.

The rate at which proteins unfold as well as the amount of the protein material sequestered in the form of NF-PPIs in the cytoplasm inherently depends on the environmental temperature: the higher is the environmental temperature, the greater is the total concentration of unfolded proteins and so is the amount of protein materials wasted in the form of NF-PPIs. In turn, the response of cells to the temperature upshift via chaperone upregulation is faster when the temperature increment is larger. When the temperature is switched back to its original low temperature value, the concentration of unfolded protein species subsides, so is the amount of protein materials captured by NF-PPIs. Consequently, the chaperones are downregulated back to their normal levels as it is more beneficial for cells to harness the cells’ limited recourses to increase cellular fitness rather than maintaining high chaperone concentrations when there is no stress. All these effects are well captured in our ab initio cell model. Given the fact that only foldase chaperones respond to heat shock by rescuing fitness by folding the unfolded proteins and reducing dramatically the amount of protein engaged in NF-PPIs, we conclude that the heat shock response triggered by the foldase chaperones is indeed an unfolded protein response.

Redistribution of cell’s resources upon heat shock

The foldase chaperones led to a genuine heat shock response by which the concentration of these chaperones is rapidly upregulated in a temperature dependent manner. However, we did not observe any heat shock response for the holdase chaperones. Understanding the mechanism behind the sequence of elementary events emerging from heat shock, by which the concentrations of proteins and chaperones are optimized to adapt to elevated temperatures, and later on, back to the original temperature is of particular interest. Since our fitness function induces a sizeable fitness cost for the overexpression of proteins and thus limits the cells’ resources by keeping the sum of overall protein concentrations approximately constant, the model cells redistribute their resources by optimizing the concentrations of their proteins and chaperones for a fast adaptation to new environmental temperatures. To capture the effect of redistribution of cells’ resources during heat shock, we analyzed the changes in the concentrations of proteins and chaperones. We compiled our results as correlation profiles between $C_{tot}$ and $C_{Ch}$ and present them in Fig. 6. The time evolution of concentration of different type of proteins and proteins complexes are also given in Fig. S4 for both the foldase and holdase chaperones.

Figure 6.

The scatter plots show how the chaperones and total protein concentrations change after heat shock stress for three different temperatures at (A) $C_{tot}$ vs. $Ch$ for foldase chaperones and (B) $C_{tot}$ vs. $Ch$ for holdase chaperones. To see this figure in color, go online.

Fig. 6 A shows strong temperature dependent correlations between $C_{tot}$ and $C_{Ch}$ for the active model. At high temperatures, the total protein concentration decreases as the concentration of foldase chaperones increases. Indeed, this effect becomes more pronounced as temperature increases. However, we do not observe any correlation between $C_{tot}$ and $C_{Ch}$ at all temperatures regimes for the passive model.

To closely monitor how cells’ resources are redistributed for different types of proteins and protein complexes, we analyzed the time evolution of monomer, heterodimers, and date triangle proteins in Fig. S4. For the active model, Fig. S4 shows that the concentrations of all types of proteins are downregulated at elevated temperatures during when the foldase chaperones are upregulated. Especially, we see that the most pronounced decrease is in the concentration of the most abundant protein, i.e., the monomer. Although we did not observe any increase in the concentration of holdase chaperones as a response to heat shock, Fig. S4 D shows a temperature dependent rapid decrease in the concentrations of monomers after the temperature jump. Furthermore, we also detect a small temperature dependent rapid decrease in the concentration of heterodimer and date triangle proteins as seen in Fig. S4, E and F. Hence, the cells equipped with holdase chaperones respond to the increase in the environmental temperature by redistributing the concentrations of their essential proteins rather than employing the holdase chaperones since this mechanism is apparently more beneficial for cells as a fast response to rescue the cellular fitness. When the temperature is dropped to its initial value, we again observe the effect of redistribution of cells’ resources and thus protein concentrations are readjusted to their originally optimized values for low temperature environment. To sum up, it appears that the holdase chaperones are neither capable of rescuing the cellular fitness under heat shock stress as our current results indicate nor they supply cells with any evolutionary advantage in a long run as shown in our previous work (22).

Discussion

The studies of population genetics as well as phenomenological approaches to molecular evolution provide the mathematical framework for better understanding of variations in genome sequences in cell populations of organisms over time. However, most of these studies assume certain a priori distribution of fitness effects of mutations or make other dramatic assumptions such that one specific genotype is more fit than others (40). In contrast to traditional approaches, in this study we postulated a physically motivated genotype-phenotype map and developed a physics-based multiscale ab initio microscopic model of living cells for population dynamics simulations to investigate the role of chaperones in adaptive evolution. The model implicitly takes into account protein homeostasis by imposing a global constraint on the total concentration of proteins in the proteomes of model cells.

The heat shock response proceeds via a complex cascade of molecular events that lead to preferential over expression of heat shock chaperones as well as other proteins such as sensors, proteases, signaling and regulatory enzymes, and transcription factors. Heat shock response in E. coli involves transient overexpression of ∼20 gene products induced by the heat-shock factor ${\sigma }_{32}$ (24,41,42). Our ab initio model does not obviously capture the full complexity of these processes. Hence, there are important differences between our cell model and real cells. For example, in our model the heat shock response is selected as a stress response to the elevated temperatures via inheritable fluctuations of abundances of cell proteins. However, real cells detect the presence of unfolded proteins in cytoplasm, which triggers a cascade of specific events at transcription and translation levels. Nevertheless, the key finding of our study is that the foldase chaperones are overexpressed in response to the increases in concentrations of unfolded proteins upon heat-shock, which is in a broad agreement with experimental results (42,43) despite the differences in mechanistic details between our ab initio cell model and real cells. The overexpression of foldase chaperones in response to heat shock is remarkably robust, as it is observed in each evolutionary trajectory. In contrast to the foldases, the holdase chaperones did not provide cells with any apparent fitness advantage upon heat shock.

Here we employed LMA to determine equilibrium concentrations of proteins in the model cytoplasm. The LMA is a mean-field equilibrium approximation, which is valid on intermediate timescales that include multiple molecular scale binding and unbinding events (44). Therefore, our formulation of chaperone actions is fundamentally coarse-grained, which does not allow to distinguish between different mechanistic proposals for active chaperone action such as active folding within a cavity (27,45) or active unfolding with subsequent kinetic partitioning of misfolded species (46–49). Our model makes a key distinction between the effective outcomes of multiple molecular events: foldase chaperones that usually consume ATP effectively increase the concentration of folded proteins, whereas holdase chaperones that do not usually consume ATP do not change the equilibrium distribution between folded and unfolded proteins. We assume that energy supply is not a limiting factor and therefore do not model the ATP consumption explicitly. The support for this energy neutrality assumption comes from the finding that the overexpression of GroEL in wild-type E. coli cells does not incur fitness cost (23).

Our model does not include irreversible effects such as aggregation and/or proteolysis. Holdase chaperones in our model are allowed to form unfolded protein-holdase complexes so as to minimize fitness cost by limiting sequestration of other proteins into NF-PPIs with the unfolded clients of holdase. A possible reason why we did not observe a fitness advantage for holdase chaperons might be that we treated the interactions of unfolded proteins with holdases as fully reversible interactions without interference of irreversible events such as aggregation and/or proteolysis. An immediate extension of our model to include these irreversible effects might reveal potential fitness advantages of holdase chaperones. However, this is not obvious or even straightforward to implement. The effective role of holdases in our model is to decrease the concentration of free unfolded proteins in a cell and thus to diminish their propensity to aggregate. A similar effect is achieved by Lon protease in E. coli (23). On the other hand, the feedback loops can cause significant shifts in protein abundances providing another potential line of defense against aggregation in heat-shocked cells (19). Quantitative multiscale modeling of combination of all these effects is nontrivial because their relative role is unknown. Therefore, it is not also clear whether holdases would provide fitness advantage upon heat-shock in real cells even in the presence of irreversible aggregation.

It is noteworthy that real cells seem to have both foldase and holdase chaperones. If foldase and holdase chaperones function on the same pathway the role of holdase would be to keep a troubled protein in a soluble complex until it encounters a foldase chaperone. A recent proteomics study (50) showed the existence of such collaboration in E. coli chaperone network in which one predominant function of DnaK was to sequester client proteins for their subsequent folding for GroEL.

Conclusions

In conclusion, our results provide an insight into how the physical principles of protein folding and interactions determine the fundamental aspects of complex biological behavior such as heat-shock response. The dynamics of heat-shock response in our minimalistic model is apparently simpler than much more complex mechanisms acting in bacterial cells. However, similarities between our findings and fundamental aspects of heat-shock response in living cells point out to protein biophysics as a fundamental factor in sculpting evolutionary dynamics of adaptation. Biophysics determines possible scenarios for cellular response to heat shock on multiple scales whereas cells evolve robust and sometimes complex mechanisms to realize them.