Evolution of sparsity and modularity in a model of protein allostery

The sequence of a protein is not only constrained by its physical and biochemical properties under current selection, but also by features of its past evolutionary history. Understanding the extent and the form that these evolutionary constraints may take is important to interpret the information in protein sequences. To study this problem, we introduce a simple but physical model of protein evolution where selection targets allostery, the functional coupling of distal sites on protein surfaces. This model shows how the geometrical organization of couplings between amino acids within a protein structure can depend crucially on its evolutionary history. In particular, two scenarios are found to generate a spatial concentration of functional constraints: high mutation rates and fluctuating selective pressures. This second scenario offers a plausible explanation for the high tolerance of natural proteins to mutations and for the spatial organization of their least tolerant amino acids, as revealed by sequence analysis and mutagenesis experiments. It also implies a faculty to adapt to new selective pressures that is consistent with observations. The model illustrates how several independent functional modules may emerge within the same protein structure, depending on the nature of past environmental fluctuations. Our model thus relates the evolutionary history of proteins to the geometry of their functional constraints, with implications for decoding and engineering protein sequences.

Figure 1

(a) A model of protein allostery is defined as a spin glass on a cylindrical lattice, a geometry chosen because of its structural homogeneity. In this model, spins ${\sigma }_i$ are on the nodes to represent physical variables, and couplings $J_{ij}$ are on the edges to represent interactions between these variables. These $J_{ij}$ are subject to evolution. Interactions with a modulator $m$ and/or a ligand $\ell$ are modeled by fields $h_i$ applied to the sites $i$ at the open ends of the cylinder. Allostery is quantified by $\varphi (J,\ell ,m)$, the difference between the binding free energy of $\ell$ in the presence of $m$, $\Delta F(J,\ell |m)$, and in its absence, $\Delta F(J,\ell |0)$. (b) The sequences of the modulator and ligand are defined by the values and signs of the fields $h_i$, with $h_i=0$ representing an interaction with the solvent. Evolution is performed over a population of systems with selection for allosteric efficiency and with mutations affecting the couplings $J_{ij}$ at a rate $\mu$. A given generation is selected for allostery with given $\ell ,m$, but when the environment fluctuates, different generations may experience different $\ell ,m$. By symmetry, varying $\ell$ or $m$ is equivalent, and we fix the sequence of the ligand to $h_i=+1$ for all $i$ at the bottom of the cylinder when $\ell$ is present and vary only the sequence of the modulator every $\tau /2$ generations, between $h_i=+1$ for $i$ at the top ($m=+$) and $h_i=-1\left(m=-\right)$. Modulators or ligands with nonuniform sequences also can be considered as in Fig. 9.

Figure 2

Examples of systems obtained from an evolutionary dynamics with mutation rate $\mu =5\times {10}^{-5}$ and different periods $\tau$ of fluctuations of selective pressure ($\tau =200,400,1000$). (a) Couplings $J_{ij}$ with large absolute values, $|J_{ij}|>0.8$. (b) Couplings $J_{ij}$ inducing a large loss in allosteric efficiency when mutated, $\delta {\varphi }_{ij}>0.1$ (see Fig. 3 for other values of the cutoffs). The figures display the fittest individual in a population of $P=500$ individuals prior to a change of environment. (c) Sparsity of evolved systems as a function of the period $\tau$ of environmental changes, where sparsity is defined as the fraction of nonrepresented couplings in (b). The error-bars (standard deviations over 10 simulations) are smaller than the marker.

Figure 3

For the system associated with $\tau =200$ in Fig. 2, couplings $|J_{ij}|$ and functional constraints $\delta {\varphi }_{ij}$ above different values of the cutoffs (Fig. 2 corresponds to the middle column, $|J_{ij}|>0.8$ and $\delta {\varphi }_{ij}>0.1$).

Figure 4

Overlap between functionally significant couplings ($\delta {\varphi }_{ij}>0.1$) between a system at generation $t_0$ and a system at generation $t_0+t$ along a same evolutionary trajectory as a function of the number of generations (time), counted in number of periods $\tau$ (the error bars are standard deviations over 100 simulations). The locations of the sectors shown in Fig. 2 are found to be stable over multiple periods of environmental changes.

Figure 5

Analysis of coevolution. To generate a data set of phylogenetically related systems analogous to the alignments of protein families used to infer sectors from protein sequences [4], we start from a population obtained by evolution under fluctuating selective pressures and then generate independent trajectories, under the same evolutionary parameters. (a) A system from the initial population, here obtained with $\tau =200$ and $\mu =5\times {10}^{-5}$, represented as Fig. 2. (b) Matrix of covariance between couplings, ${\mathcal{C}}_{ij,kl}=|\langle J_{ij}{J}_{kl}\rangle -\langle J_{ij}\rangle \langle J_{kl}\rangle |$, computed from the results of 100 independent trajectories run over $3\tau =600$ generations for the same values of the parameters $\mu ,\tau$; $\langle ...\rangle$ denotes an average over the different populations (the absolute value is taken to treat equivalently positive and negative covariations). The couplings are ordered based on the first eigenvector (principal component) of the matrix. (c) The top positions along this principal component define a sector, which overlaps with the sector defined in (a) for the original system.

Figure 6

(a) Sparsity of evolved proteins as a function of the mutation rate $\mu$ and the period $\tau$ of fluctuating selective pressures. Sparsity is defined as the fraction of couplings with $\delta {\varphi }_{ij}<0.1$ [nonrepresented couplings in Fig. 2]. Below the dashed line, the environmental fluctuations are too fast for the population to follow them, and the systems are nonadapted (region $na$). Sparse systems are found in two ranges of parameters: for intermediate values of $\mu \tau$, where they are driven by fluctuating selective pressures (region $F$, including the three systems of Fig. 2 indicated by crosses), and for high values of $\mu$, where they are driven by a large mutational load (region $M$). (b) Fitness of the population as a function of $\mu$ and $\tau$; the axis and the red line are the same as in (a). The dashed line ($\varphi ={10}^{-7}$) corresponds to the typical maximal value of allosteric efficiency in populations of $P=500$ random systems. Below this line, the populations may be considered as nonadapted.

Figure 7

Sparsity as a function of the scaling variable $\tau \mu P$ for systems obtained from evolutionary dynamics with different values of the period $\tau$ of environmental changes and mutation rate $\mu$ (in the range of values shown in Fig. 6) and for two population sizes, $P=100$, 500.

Figure 8

(a) Robustness of evolved systems as a function of the period $\tau$ of environmental changes. (b) Evolvability. The error bars are standard deviations over 10 simulations.

Figure 9

(a) Sparsity as a function of the sequence similarity $s$ between the two alternative modulator sequences, for $\mu ={10}^{-5}$ and $\tau =100$ (mean and standard deviation over 10 simulations). (b) Minimal number of point mutations necessary to adapt to a new random modulator as a function of sparsity. The simulations are obtained with different values of $\tau \in [10,...,5000]$ and $\mu \in [5\times {10}^{-3},...,2\times {10}^{-6}]$ corresponding to adapted populations ($\varphi >{10}^{-7}$); the results are averages over five random modulators.

Figure 10

(a) Sparsity as a function of allosteric efficiency (fitness) for evolved systems obtained with various $\mu <{10}^{-2}$ and $\tau >10$ (blue diamond) and for typical systems with same fitness obtained by Monte Carlo sampling (red circle). (b) Distance to a system with different function (i.e., allostery induced by a modulator $m^{\prime }$ different from the one $m$ for which the system was last selected), measured by the minimal number of beneficial mutations needed to reach an equivalent allosteric efficiency after the change $m\to m^{\prime }$, for evolved systems (blue diamond) and typical systems (red circle). Systems evolved in a fluctuating environment are thus atypical amongst systems with equivalent fitness value for being sparser and closer to solutions to new selective challenges.

Figure 11

Typical outcomes for a variant of the model where the regulatory site is partitioned into two subsites, each associated with an independently varying modulator, and where selection is on allostery in the presence of at least one of the two modulators $m_1$ and/or $m_2$. Shown are the couplings with $\delta {\varphi }_{ij}>0.1$ as in Fig. 2. (a) A nonmodular system, with a single sector localized at the interface between the two regulatory sites. (b) A modular system, with two distinct sectors. These two systems are the (stable) outcomes of two distinct evolutionary trajectories with same evolutionary parameters ($\tau =100$, $\mu =5\times {10}^{-4}$). The difference stems only from stochastic effects. The table indicates the probability to obtain a modular system for two values of the period $\tau$ and for $m_1,m_2$ varying either simultaneously or consecutively [49].

Figure 12

Sparsity as a function of the time scale $\tau$ of environmental changes for systems that evolved subject to different mutational processes: (a) Identical to Fig. 2: the $J_{ij}$ are mutated to a random value uniformly distributed in $[-1,1]$, independently of their previous value. (b) The $J_{ij}$ are drawn from a finite set of discrete values. (c) Sum rule with variance ${\sigma }_s^2=0.2$. (d) Product rule with variance ${\sigma }_p^2=1.6$. In each case, the sparsity tends to decrease with increasing values of the period $\tau$ of the environmental changes.