Bistable biochemical switches are widely found in gene regulatory networks and signal transduction pathways. Their switching dynamics are difficult to study, however, because switching events are rare, and the systems are out of equilibrium. We present a simulation method for predicting the rate and mechanism of the flipping of these switches. We apply it to a genetic switch and find that it is highly efficient. The path ensembles for the forward and reverse processes do not coincide. The method is widely applicable to rare events and nonequilibrium processes.

We present three algorithms for calculating rate constants and sampling transition paths for rare events in simulations with stochastic dynamics. The methods do not require a priori knowledge of the phase-space density and are suitable for equilibrium or nonequilibrium systems in stationary state. All the methods use a series of interfaces in phase space, between the initial and final states, to generate transition paths as chains of connected partial paths, in a ratchetlike manner. No assumptions are made about the distribution of paths at the interfaces. The three methods differ in the way that the transition path ensemble is generated. We apply the algorithms to kinetic Monte Carlo simulations of a genetic switch and to Langevin dynamics simulations of intermittently driven polymer translocation through a pore. We find that the three methods are all of comparable efficiency, and that all the methods are much more efficient than brute-force simulation.

Multisite covalent modification of proteins is omnipresent in eukaryotic cells. A well-known example is the mitogen-activated protein kinase (MAPK) cascade where, in each layer of the cascade, a protein is phosphorylated at two sites. It has long been known that the response of a MAPK pathway strongly depends on whether the enzymes that modify the protein act processively or distributively. A distributive mechanism, in which the enzyme molecules have to release the substrate molecules in between the modification of the two sites, can generate an ultrasensitive response and lead to hysteresis and bistability. We study by Green’s Function Reaction Dynamics (GFRD), a stochastic scheme that makes it possible to simulate biochemical networks at the particle level in time and space, a dual phosphorylation cycle in which the enzymes act according to a distributive mechanism. We find that the response of this network can differ dramatically from that predicted by a mean-field analysis based on the chemical rate equations. In particular, rapid rebindings of the enzyme molecules to the substrate molecules after modification of the first site can markedly speed up the response and lead to loss of ultrasensitivity and bistability. In essence, rapid enzyme-substrate rebindings can turn a distributive mechanism into a processive mechanism. We argue that slow ADP release by the enzymes can protect the system against these rapid rebindings, thus enabling ultrasensitivity and bistability.

Significance: Cells continually have to sense their environments to make decisions—to stay put or move, to differentiate or proliferate, or even to live or die. However, they are thwarted by noise at the cellular scale. Cells use signaling networks to filter this noise as much as possible and sense accurately. To operate these networks, resources are required: time, protein copies, and energy. We present a theory for the optimal design of cellular sensing systems that maximize sensing precision given these resources. It reveals a new design principle, namely that of optimal resource allocation. It describes how these resources must be allocated so that none are wasted. We show that the chemotaxis network of Escherichia coli obeys this principle.

In a recent series of ground-breaking experiments, Nakajima et al. [Nakajima M, Imai K, Ito H, Nishiwaki T, Murayama Y, Iwasaki H, Oyama T, Kondo T (2005) Science 308:414–415] showed that the three cyanobacterial clock proteins KaiA, KaiB, and KaiC are sufficient in vitro to generate circadian phosphorylation of KaiC. Here, we present a mathematical model of the Kai system. At its heart is the assumption that KaiC can exist in two conformational states, one favoring phosphorylation and the other dephosphorylation. Each individual KaiC hexamer then has a propensity to be phosphorylated in a cyclic manner. To generate macroscopic oscillations, however, the phosphorylation cycles of the different hexamers must be synchronized. We propose a novel synchronization mechanism based on differential affinity: KaiA stimulates KaiC phosphorylation, but the limited supply of KaiA dimers binds preferentially to those KaiC hexamers that are falling behind in the oscillation. KaiB sequesters KaiA and stabilizes the dephosphorylating KaiC state. We show that our model can reproduce a wide range of published data, including the observed insensitivity of the oscillation period to variations in temperature, and that it makes nontrivial predictions about the effects of varying the concentrations of the Kai proteins.