Exploratory adaptation in large random networks

Abstract

The capacity of cells and organisms to respond to challenging conditions in a repeatable manner is limited by a finite repertoire of pre-evolved adaptive responses. Beyond this capacity, cells can use exploratory dynamics to cope with a much broader array of conditions. However, the process of adaptation by exploratory dynamics within the lifetime of a cell is not well understood. Here we demonstrate the feasibility of exploratory adaptation in a high-dimensional network model of gene regulation. Exploration is initiated by failure to comply with a constraint and is implemented by random sampling of network configurations. It ceases if and when the network reaches a stable state satisfying the constraint. We find that successful convergence (adaptation) in high dimensions requires outgoing network hubs and is enhanced by their auto-regulation. The ability of these empirically validated features of gene regulatory networks to support exploratory adaptation without fine-tuning, makes it plausible for biological implementation.

Figure 1. Exploratory dynamics and convergence to a constraint-satisfying stable state.

(a) Schematic representation of the model: a random N × N network, composed of an adjacency matrix T and an interaction strength matrix J, governs a nonlinear dynamical system (equation in box; φ(x)=tanh(x)). The resulting spontaneous dynamics are typically irregular for large enough interactions. A macroscopic variable, the phenotype y, is subject to an arbitrary constraint y≈y* with finite precision ɛ. When the constraint is not met (left; ‘hot' regime), the connections strengths J_ij undergo a random walk with magnitude determined by the coefficient D and the mismatch function (y−y*). The random walk stops when the mismatch is stably reduced to zero (right; ‘frozen' regime). (b–d) Example of exploration and convergence. Shown are representative trajectories of connection strengths (b), microscopic variables (c) and the phenotype y (d) before and after convergence to a stable state satisfying the constraint. The network in this example has scale-free (SF) out-degree distribution (a=1, γ=2.4) and Binomial in-degree distribution . N=1,000, y*=10, D=10⁻³, g₀=10. See Methods for more details.

Figure 2. Convergence fractions depend on network topology.

(a) Seven ensembles of networks of size N=1,500 and different topologies exhibit remarkably different convergence fractions (CFs). Ensembles are characterized by the out- and in- degree distributions of the adjacency matrix T: ‘SF', scale free distribution; ‘Exp', exponential distribution; ‘Binom', Binomial distribution. (b) CF as a function of network size for the same ensembles of (a) with matching colours. N=1,500, y=0, g₀=10, D=10⁻³. Parameters for degree distributions: SF, (a=1, γ=2.4); Binom, ; Exp, (β=3.5).

Figure 3. Exploratory adaptation depends on the existence of hubs and is enhanced by their auto-regulation.

(a) CF versus out-degree of the largest hub in a collection of SF-Binom networks binned according to their largest hub. (b) Changes in Convergence Fraction (CF) following deletion of a number of leading hubs (red) or deletion of the same number of random nodes (blue) from networks with SF-Binom topology. (c) Effect of adding a small number of outgoing hubs to a Binon-Binom ensemble. The out-degrees of the added hubs were chosen to mimic the SF-out ensemble of Fig. 2. (d) Effect of adding autoregulatory loops on a specific number (1, 3 and 10) of the leading outgoing hubs on a background of a SF-Binom ensemble. N=1,500, y=0, g₀=10, D=10⁻³. Parameters for degree distributions: SF, (a=1, γ=2.4); Binom, ; Exp, (β=3.5).

Figure 4. Effect of positive autoregulation on convergence fractions.

Positive autoregulatory loops were added randomly to 10% of the nodes in four ensembles, each comprising 500 networks of a given size (N=1,000 or 3,000) and topology (SF-Binom or vice versa). Convergence in each ensemble is compared to controls without extra loops, with and without matching of the degree distributions to the enriched ensemble (Null 1 and Null2, respectively). Parameters of the SF and Binom distributions (prior to addition of loops) are: SF, (a=1, γ=2.4) and Binomial, (p= and N). Other parameters are g₀=10, α=100, ɛ=3, c=0.2, D=10⁻³ and y=0.

Figure 5. Dependence of CF on model parameters.

(a) CF versus ɛ, the width of the comfort zone around y*. (b) CF (blue) versus the constraint value y. For comparison, the grey curve shows the fraction of time in which y(t) spontaneously reaches the constraint-satisfying range. (c) CF versus the strength of exploratory random walk in connection strengths, D. (d) CF versus g₀ (proportional to the s.d. of connection strengths; see Methods for details). Network ensemble with SF-out (a=1, γ=2.4) and Binom-in degree distributions. Unless otherwise specified, all ensembles have N=1,000, y*=0, g₀=10, and D=10⁻³.

Figure 6. Distribution of convergence times for networks which converged in less than 10⁴ timesteps.

Solid lines depict stretched exponential fits. (a) Probability density distribution (PDF) of convergence time for three topological ensembles. (b) PDFs after deleting the 8 largest hubs (red) or the same number of randomly-chosen nodes (light blue) from the SF-Binom ensemble. All ensembles have N=1,000, y=0, g₀=10, and D=10⁻³. Degree distribution parameters: SF, (a=1, γ=2.4); Binom, ; Exp, ().

Figure 7. Fixed points in the absence of exploration for different topological ensembles.

(a) Fraction of networks within an ensemble which relaxed to a fixed point under the nonlinear dynamics of equation (1), with fixed connections, no constraint and no feedback. Topological ensembles which exhibited higher success in exploratory adaptation in Fig. 2b, relaxed to fixed points in a larger fraction of simulations. (b) Distribution of relaxation times to fixed points for two of the ensembles. Note the shorter typical timescale for the SF-Exp ensemble (the more successfully adapting ensemble). N=1,000, g₀=10. Dergree distribution parameters: SF, (a=1, γ=2.4); Binom, ; Exp, ().