Julian B. Voits

For irreducible, time-homogeneous Markov networks, mutual linearity has recently been established for both occupation probabilities and network currents in the stationary regime as well as in the non-stationary regime in Laplace space. The derivation of this property for the stationary distribution utilized the Markov chain tree theorem, which also allows for an explicit combinatorial expression of the response ratios under variation of a single transition rate. The extension of this result was proven at the trajectory level by employing the Doob–Meyer decomposition. By employing the all-minors matrix-tree theorem, we show that this property also follows from a graph theoretic formulation and derive explicit combinatorial expressions for the non-stationary response ratios. The stationary result follows as the long-time limit. We also show that the small-time asymptotics are entirely determined by minimal path distances in the underlying graph. Finally we use the graph theoretic approach to prove that mutual linearity also extends to hitting time densities.

First-passage-times are often the most relevant aspect of a complex Markovian network because they signify when information processing has resulted in a definite decision. Previous studies have shown that for kinetic proofreading networks in the limit of large network size the first-passage-time distribution converges either to a \ensuremath{\delta} or to an exponential distribution. Remarkably, these two forms correspond to the two extreme distributions of minimal and maximal entropy for a fixed mean, respectively. Here we build on the connection between first-passage-times and graph theory to show that these two limits are not model-specific, but arise generically in Markovian networks from the distribution of the eigenvalues of the generator matrix. A deterministic peak emerges when infinitely many eigenvalues contribute, while the exponential limit arises from a single dominant eigenvalue. We also show that the exponential limit emerges robustly for reversible networks when the mean first-passage-time from the initial state to the target state becomes much larger than the mean first-passage-time in the reverse direction. In contrast, the deterministic limit is not obtained from a simple reversal of this condition, but follows from a nonvanishing conductance or a mean-residual lifetime of the process which becomes small compared to the mean first-passage-time in the long-time limit. This reveals a fundamental asymmetry between the two regimes. Our theoretical analysis is illustrated and validated by computer simulations of one-step master equations and random networks.

Vesicle-mediated secretion of ions or molecules is a central mechanism of cellular communication, for example in processes such as neurotransmission or hormone release. These events are inherently stochastic: vesicle fusions lead to bursts of variable sizes, releasing discrete packets of transmitters that are subsequently cleared or degraded. The dynamics are intrinsically time-directed due to the interplay of spontaneous bursts and continuous degradation. Using generating functions and a recursion relation, we derive an exact solution for the full time-dependent probability distribution of a general batch arrival-degradation model. This framework also enables a full analysis of first-passage times to a concentration threshold representing downstream activation. We show that activation kinetics are not determined by mean dynamics alone, but depend sensitively on the temporal statistics of arrival events, batch-size variability, and degradation. In particular, different arrival processes with identical mean rates can lead to qualitatively distinct first-passage behavior, reflecting the role of time-asymmetric fluctuations. We also discuss extensions incorporating vesicle depletion. Our results provide a transparent link between stochastic release dynamics and activation timing in vesicle-mediated signaling.

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 such as 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. We also discuss under which conditions this generic response can be violated, for example for linear chains, which have only one spanning tree.