Sensitivity Amplification in the Phosphorylation-Dephosphorylation Cycle: Nonequilibrium steady states, chemical master equation and temporal cooperativity

A new type of cooperativity termed temporal cooperativity [Biophys. Chem. 105 585-593 (2003), Annu. Rev. Phys. Chem. 58 113-142 (2007)], emerges in the signal transduction module of phosphorylation-dephosphorylation cycle (PdPC). It utilizes multiple kinetic cycles in time, in contrast to allosteric cooperativity that utilizes multiple subunits in a protein. In the present paper, we thoroughly investigate both the deterministic (microscopic) and stochastic (mesoscopic) models, and focus on the identification of the source of temporal cooperativity via comparing with allosteric cooperativity. A thermodynamic analysis confirms again the claim that the chemical equilibrium state exists if and only if the phosphorylation potential $\triangle G=0$, in which case the amplification of sensitivity is completely abolished. Then we provide comprehensive theoretical and numerical analysis with the first-order and zero-order assumptions in phosphorylation-dephosphorylation cycle respectively. Furthermore, it is interestingly found that the underlying mathematics of temporal cooperativity and allosteric cooperativity are equivalent, and both of them can be expressed by "dissociation constants", which also characterizes the essential differences between the simple and ultrasensitive PdPC switches. Nevertheless, the degree of allosteric cooperativity is restricted by the total number of sites in a single enzyme molecule which can not be freely regulated, while temporal cooperativity is only restricted by the total number of molecules of the target protein which can be regulated in a wide range and gives rise to the ultrasensitivity phenomenon.Comment: 42 pages, 13 figure

The stochastic behavior of a molecular switching circuit with feedback

Background: Using a statistical physics approach, we study the stochastic switching behavior of a model circuit of multisite phosphorylation and dephosphorylation with feedback. The circuit consists of a kinase and phosphatase acting on multiple sites of a substrate that, contingent on its modification state, catalyzes its own phosphorylation and, in a symmetric scenario, dephosphorylation. The symmetric case is viewed as a cartoon of conflicting feedback that could result from antagonistic pathways impinging on the state of a shared component. Results: Multisite phosphorylation is sufficient for bistable behavior under feedback even when catalysis is linear in substrate concentration, which is the case we consider. We compute the phase diagram, fluctuation spectrum and large-deviation properties related to switch memory within a statistical mechanics framework. Bistability occurs as either a first-order or second-order non-equilibrium phase transition, depending on the network symmetries and the ratio of phosphatase to kinase numbers. In the second-order case, the circuit never leaves the bistable regime upon increasing the number of substrate molecules at constant kinase to phosphatase ratio. Conclusions: The number of substrate molecules is a key parameter controlling both the onset of the bistable regime, fluctuation intensity, and the residence time in a switched state. The relevance of the concept of memory depends on the degree of switch symmetry, as memory presupposes information to be remembered, which is highest for equal residence times in the switched states. Reviewers: This article was reviewed by Artem Novozhilov (nominated by Eugene Koonin), Sergei Maslov, and Ned Wingreen.Comment: Version published in Biology Direct including reviewer comments and author responses, 28 pages, 7 figure

Model reduction for stochastic CaMKII reaction kinetics in synapses by graph-constrained correlation dynamics.

A stochastic reaction network model of Ca(2+) dynamics in synapses (Pepke et al PLoS Comput. Biol. 6 e1000675) is expressed and simulated using rule-based reaction modeling notation in dynamical grammars and in MCell. The model tracks the response of calmodulin and CaMKII to calcium influx in synapses. Data from numerically intensive simulations is used to train a reduced model that, out of sample, correctly predicts the evolution of interaction parameters characterizing the instantaneous probability distribution over molecular states in the much larger fine-scale models. The novel model reduction method, 'graph-constrained correlation dynamics', requires a graph of plausible state variables and interactions as input. It parametrically optimizes a set of constant coefficients appearing in differential equations governing the time-varying interaction parameters that determine all correlations between variables in the reduced model at any time slice

Enzyme sequestration by the substrate: An analysis in the deterministic and stochastic domains.

This paper is concerned with the potential multistability of protein concentrations in the cell. That is, situations where one, or a family of, proteins may sit at one of two or more different steady state concentrations in otherwise identical cells, and in spite of them being in the same environment. For models of multisite protein phosphorylation for example, in the presence of excess substrate, it has been shown that the achievable number of stable steady states can increase linearly with the number of phosphosites available. In this paper, we analyse the consequences of adding enzyme docking to these and similar models, with the resultant sequestration of phosphatase and kinase by the fully unphosphorylated and by the fully phosphorylated substrates respectively. In the large molecule numbers limit, where deterministic analysis is applicable, we prove that there are always values for these rates of sequestration which, when exceeded, limit the extent of multistability. For the models considered here, these numbers are much smaller than the affinity of the enzymes to the substrate when it is in a modifiable state. As substrate enzyme-sequestration is increased, we further prove that the number of steady states will inevitably be reduced to one. For smaller molecule numbers a stochastic analysis is more appropriate, where multistability in the large molecule numbers limit can manifest itself as multimodality of the probability distribution; the system spending periods of time in the vicinity of one mode before jumping to another. Here, we find that substrate enzyme sequestration can induce bimodality even in systems where only a single steady state can exist at large numbers. To facilitate this analysis, we develop a weakly chained diagonally dominant M-matrix formulation of the Chemical Master Equation, allowing greater insights in the way particular mechanisms, like enzyme sequestration, can shape probability distributions and therefore exhibit different behaviour across different regimes

Robust circadian clocks from coupled protein modification and transcription-translation cycles

The cyanobacterium Synechococcus elongatus uses both a protein phosphorylation cycle and a transcription-translation cycle to generate circadian rhythms that are highly robust against biochemical noise. We use stochastic simulations to analyze how these cycles interact to generate stable rhythms in growing, dividing cells. We find that a protein phosphorylation cycle by itself is robust when protein turnover is low. For high decay or dilution rates (and co mpensating synthesis rate), however, the phosphorylation-based oscillator loses its integrity. Circadian rhythms thus cannot be generated with a phosphorylation cycle alone when the growth rate, and consequently the rate of protein dilution, is high enough; in practice, a purely post-translational clock ceases to function well when the cell doubling time drops below the 24 hour clock period. At higher growth rates, a transcription-translation cycle becomes essential for generating robust circadian rhythms. Interestingly, while a transcription-translation cycle is necessary to sustain a phosphorylation cycle at high growth rates, a phosphorylation cycle can dramatically enhance the robustness of a transcription-translation cycle at lower protein decay or dilution rates. Our analysis thus predicts that both cycles are required to generate robust circadian rhythms over the full range of growth conditions.Comment: main text: 7 pages including 5 figures, supplementary information: 13 pages including 9 figure

Simplicity of Completion Time Distributions for Common Complex Biochemical Processes

Biochemical processes typically involve huge numbers of individual reversible steps, each with its own dynamical rate constants. For example, kinetic proofreading processes rely upon numerous sequential reactions in order to guarantee the precise construction of specific macromolecules. In this work, we study the transient properties of such systems and fully characterize their first passage (completion) time distributions. In particular, we provide explicit expressions for the mean and the variance of the completion time for a kinetic proofreading process and computational analyses for more complicated biochemical systems. We find that, for a wide range of parameters, as the system size grows, the completion time behavior simplifies: it becomes either deterministic or exponentially distributed, with a very narrow transition between the two regimes. In both regimes, the dynamical complexity of the full system is trivial compared to its apparent structural complexity. Similar simplicity is likely to arise in the dynamics of many complex multi-step biochemical processes. In particular, these findings suggest not only that one may not be able to understand individual elementary reactions from macroscopic observations, but also that such understanding may be unnecessary

Phase resetting reveals network dynamics underlying a bacterial cell cycle

Genomic and proteomic methods yield networks of biological regulatory interactions but do not provide direct insight into how those interactions are organized into functional modules, or how information flows from one module to another. In this work we introduce an approach that provides this complementary information and apply it to the bacterium Caulobacter crescentus, a paradigm for cell-cycle control. Operationally, we use an inducible promoter to express the essential transcriptional regulatory gene ctrA in a periodic, pulsed fashion. This chemical perturbation causes the population of cells to divide synchronously, and we use the resulting advance or delay of the division times of single cells to construct a phase resetting curve. We find that delay is strongly favored over advance. This finding is surprising since it does not follow from the temporal expression profile of CtrA and, in turn, simulations of existing network models. We propose a phenomenological model that suggests that the cell-cycle network comprises two distinct functional modules that oscillate autonomously and couple in a highly asymmetric fashion. These features collectively provide a new mechanism for tight temporal control of the cell cycle in C. crescentus. We discuss how the procedure can serve as the basis for a general approach for probing network dynamics, which we term chemical perturbation spectroscopy (CPS)