The generic temperature response of large biochemical networks

TL;DR

The paper resolves why large biochemical networks exhibit a quadratic Arrhenius response to temperature by framing development as a mean first-passage time problem on a graph. It derives a general expression for the MFPT in terms of spanning trees and two-tree forests, and shows that, under a large-system limit with Gaussian-distributed activation energies, the log-rate is dominated by the first two cumulants, yielding $ ext{ln}ig\uparrow auigig o (ig angle Eig angle_T - ig angle Eig angle_F) ablaeta + frac{1}{2}( ext{Var}_F - ext{Var}_T) ablaeta^2 + ext{const}$, i.e., a quadratic Arrhenius plot. This theory, supported by simulations and experimental developmental data from flies and frogs, explicates a generic temperature dependence for large networks and clarifies deviations arising in small networks, linear chains, or correlated energies. The approach provides a scalable method to estimate MFPTs without full system-resolution and offers a principled link between network architecture, energy statistics, and temperature sensitivity with implications for development and climate-related biology.

Abstract

Biological systems are remarkably susceptible to relatively small temperature changes. The most obvious example is fever, when a modest rise in body temperature of only few Kelvin has strong effects on our immune system and how it fights pathogens. Another very important example is climate change, when even smaller temperature changes lead to dramatic shifts in ecosystems. Although it is generally accepted that the main effect of an increase in temperature is the acceleration of biochemical reactions according to the Arrhenius equation, it is not clear how it affects large biochemical networks with complicated architectures. For developmental systems like fly and frog, it has been shown that the system response to temperature deviates in a characteristic manner from the linear Arrhenius plot of single reactions, but a rigorous explanation has not been given yet. Here we use a graph-theoretical interpretation of the mean first-passage times of a biochemical master equation to give a statistical description. We find that in the limit of large system size and if the network has a bias towards a target state, then the Arrhenius plot is generically quadratic, in excellent agreement with numerical simulations for large networks as well as with experimental data for developmental times in fly and frog. We also discuss under which conditions this generic response can be violated, for example for linear chains, which have only one spanning tree.

Figures (24)

• Figure 1: The Arrhenius plot depends on network size. (a) The temperature dependence of the rate of a single biochemical reaction, such as ligand-receptor binding, usually follows the Arrhenius equation, resulting in a linear relationship in the Arrhenius plot (logarithmic rates against inverse temperature). (b) For networks of intermediate size, like the MAPK-signaling pathway shown here, the temperature response is more complex and depends on microscopic details. (c) In large biochemical networks, such as those governing the development from a Drosophila fertilized egg to a larva, a quadratic dependence emerges in the Arrhenius plot.
• Figure 2: An illustration of the graph-theoretical decomposition of networks into spanning trees and forests used throughout this work. (a) The full directed graph on three vertices. (b) A spanning tree rooted at $3$. (c) A spanning forest of two trees rooted at $2$ and $3$.
• Figure 3: All spanning trees of the complete graph on four vertices rooted at $4$. The graph-theoretical interpretation of the sum in Eq. \ref{['eq:sum_spanning_trees']} consists in finding the spanning trees. The vertex labels are for all graphs as indicated for the first one.
• Figure 4: All spanning forests of the complete graph on four vertices rooted at $4$. The graph-theoretical interpretation of the summands in Eq.\ref{['eq:sum_spanning_forests']} consists in finding the two-tree spanning forests $\mathcal{F}_{[1,4]}^{1\rightarrow 1}$, $\mathcal{F}_{[2,4]}^{1\rightarrow 2}$ and $\mathcal{F}_{[3,4]}^{1\rightarrow 3}$ of the complete graph on four vertices rooted at $4$. The vertex labels are for all graphs as indicated for the first one.
• Figure 5: Randomly generated networks with features that are typical for development of sizes (a) $N_v=30$ and (b) $N_v=50$ vertices. The initial vertices are represented as squares and the final vertices as diamonds. The networks are biased towards the final vertices and contain compact elements with many connections.