Effective reaction rates in diffusion-limited phosphorylation-dephosphorylation cycles

Discretised Flux Balance Analysis for Reaction-Diffusion Simulation of Single-Cell Metabolism

Metabolites have to diffuse within the sub-cellular compartments they occupy to specific locations where enzymes are, so reactions could occur. Conventional flux balance analysis (FBA), a method based on linear programming that is commonly used to model metabolism, implicitly assumes that all enzymatic reactions are not diffusion-limited though that may not always be the case. In this work, we have developed a spatial method that implements FBA on a grid-based system, to enable the exploration of diffusion effects on metabolism. Specifically, the method discretises a living cell into a two-dimensional grid, represents the metabolic reactions in each grid element as well as the diffusion of metabolites to and from neighbouring elements, and simulates the system as a single linear programming problem. We varied the number of rows and columns in the grid to simulate different cell shapes, and the method was able to capture diffusion effects at different shapes. We then used the method to simulate heterogeneous enzyme distribution, which suggested a theoretical effect on variability at the population level. We propose the use of this method, and its future extensions, to explore how spatiotemporal organisation of sub-cellular compartments and the molecules within could affect cell behaviour.

Evolution of opposing regulatory interactions underlies the emergence of eukaryotic cell cycle checkpoints

In eukaryotes the entry into mitosis is initiated by activation of cyclin-dependent kinases (CDKs), which in turn activate a large number of protein kinases to induce all mitotic processes. The general view is that kinases are active in mitosis and phosphatases turn them off in interphase. Kinases activate each other by cross- and self-phosphorylation, while phosphatases remove these phosphate groups to inactivate kinases. Crucial exceptions to this general rule are the interphase kinase Wee1 and the mitotic phosphatase Cdc25. Together they directly control CDK in an opposite way of the general rule of mitotic phosphorylation and interphase dephosphorylation. Here we investigate why this opposite system emerged and got fixed in almost all eukaryotes. Our results show that this reversed action of a kinase-phosphatase pair, Wee1 and Cdc25, on CDK is particularly suited to establish a stable G2 phase and to add checkpoints to the cell cycle. We show that all these regulators appeared together in LECA (Last Eukaryote Common Ancestor) and co-evolved in eukaryotes, suggesting that this twist in kinase-phosphatase regulation was a crucial step happening at the emergence of eukaryotes.

A predictive model of gene expression reveals the role of network motifs in the mating response of yeast

Cells use signaling pathways to receive and process information about their environment. These nonlinear systems rely on feedback and feedforward regulation to respond appropriately to changing environmental conditions. Mathematical models describing signaling pathways often lack predictive power because they are not trained on data that encompass the diverse time scales on which these regulatory mechanisms operate. We addressed this limitation by measuring transcriptional changes induced by the mating response in Saccharomyces cerevisiae exposed to different dynamic patterns of pheromone. We found that pheromone-induced transcription persisted after pheromone removal and showed long-term adaptation upon sustained pheromone exposure. We developed a model of the regulatory network that captured both characteristics of the mating response. We fit this model to experimental data with an evolutionary algorithm and used the parameterized model to predict scenarios for which it was not trained, including different temporal stimulus profiles and genetic perturbations to pathway components. Our model allowed us to establish the role of four architectural elements of the network in regulating gene expression. These network motifs are incoherent feedforward, positive feedback, negative feedback, and repressor binding. Experimental and computational perturbations to these network motifs established a specific role for each in coordinating the mating response to persistent and dynamic stimulation.

Sampling rare events in stochastic reaction-diffusion systems within trajectory looping

In bistable reaction-diffusion systems, transitions between stable states typically occur on timescales orders of magnitude longer than the chemical equilibration time. Estimation of transition rates within explicit Brownian dynamics simulations is computationally prohibitively costly. We present a method that exploits a single trajectory, generated by a prior simulation of diffusive motions of molecules, to sample chemical kinetic processes on timescales several orders of magnitude longer than the duration of the diffusive trajectory. In this approach, we "loop" the diffusive trajectory by transferring chemical states of the molecules from the last to the first time step of the trajectory. Trajectory looping can be applied to enhance sampling of rare events in biochemical systems in which the number of reacting molecules is constant, as in cellular signal transduction pathways. As an example, we consider a bistable system of autophosphorylating kinases, for which we calculate state-to-state transition rates and traveling wave velocities. We provide an open-source implementation of the method.

SPATKIN: a simulator for rule-based modeling of biomolecular site dynamics on surfaces

Summary

Rule-based modeling is a powerful approach for studying biomolecular site dynamics. Here, we present SPATKIN, a general-purpose simulator for rule-based modeling in two spatial dimensions. The simulation algorithm is a lattice-based method that tracks Brownian motion of individual molecules and the stochastic firing of rule-defined reaction events. Because rules are used as event generators, the algorithm is network-free, meaning that it does not require to generate the complete reaction network implied by rules prior to simulation. In a simulation, each molecule (or complex of molecules) is taken to occupy a single lattice site that cannot be shared with another molecule (or complex). SPATKIN is capable of simulating a wide array of membrane-associated processes, including adsorption, desorption and crowding. Models are specified using an extension of the BioNetGen language, which allows to account for spatial features of the simulated process.

Availability and implementation

The C ++ source code for SPATKIN is distributed freely under the terms of the GNU GPLv3 license. The source code can be compiled for execution on popular platforms (Windows, Mac and Linux). An installer for 64-bit Windows and a macOS app are available. The source code and precompiled binaries are available at the SPATKIN Web site (http://pmbm.ippt.pan.pl/software/spatkin).

Contact

spatkin.simulator@gmail.com.

Supplementary information

Supplementary data are available at Bioinformatics online.

Effective reaction rates for diffusion-limited reaction cycles

Biological signals in cells are transmitted with the use of reaction cycles, such as the phosphorylation-dephosphorylation cycle, in which substrate is modified by antagonistic enzymes. An appreciable share of such reactions takes place in crowded environments of two-dimensional structures, such as plasma membrane or intracellular membranes, and is expected to be diffusion-controlled. In this work, starting from the microscopic bimolecular reaction rate constants and using estimates of the mean first-passage time for an enzyme-substrate encounter, we derive diffusion-dependent effective macroscopic reaction rate coefficients (EMRRC) for a generic reaction cycle. Each EMRRC was found to be half of the harmonic average of the microscopic rate constant (phosphorylation c or dephosphorylation d), and the effective (crowding-dependent) motility divided by a slowly decreasing logarithmic function of the sum of the enzyme concentrations. This implies that when c and d differ, the two EMRRCs scale differently with the motility, rendering the steady-state fraction of phosphorylated substrate molecules diffusion-dependent. Analytical predictions are verified using kinetic Monte Carlo simulations on the two-dimensional triangular lattice at the single-molecule resolution. It is demonstrated that the proposed formulas estimate the steady-state concentrations and effective reaction rates for different sets of microscopic reaction rates and concentrations of reactants, including a non-trivial example where with increasing diffusivity the fraction of phosphorylated substrate molecules changes from 10% to 90%.