Table of contents

Methods are presented for simulating chemical reaction networks with a spatial resolution that is accurate to nearly the size scale of individual molecules. Using an intuitive picture of chemical reaction systems, each molecule is treated as a point-like particle that diffuses freely in three-dimensional space. When a pair of reactive molecules collide, such as an enzyme and its substrate, a reaction occurs and the simulated reactants are replaced by products. Achieving accurate bimolecular reaction kinetics is surprisingly difficult, requiring a careful consideration of reaction processes that are often overlooked. This includes whether the rate of a reaction is at steady-state and the probability that multiple reaction products collide with each other to yield a back reaction. Inputs to the simulation are experimental reaction rates, diffusion coefficients and the simulation time step. From these are calculated the simulation parameters, including the 'binding radius' and the 'unbinding radius', where the former defines the separation for a molecular collision and the latter is the initial separation between a pair of reaction products. Analytic solutions are presented for some simulation parameters while others are calculated using look-up tables. Capabilities of these methods are demonstrated with simulations of a simple bimolecular reaction and the Lotka–Volterra system.

Many small globular proteins are traditionally classified as thermodynamical two-state systems, i.e., the protein is either in the native, active state (folded) or in the denatured state (unfolded). We challenge this view and show that there may exist (protein) systems for which a van't Hoff analysis of experimental data cannot determine whether the system corresponds to two or three thermodynamical states when only temperatures in a narrow temperature region around the transition are considered. We generalize a widely employed two-state protein folding model to include a third, transition state. For this three-state system we systematically study the deviation of the calorimetric enthalpy (heat of transition) from the van't Hoff enthalpy, a measure of the two-stateness of a transition. We show that under certain conditions the heat capacity of the three-state system can be almost indistinguishable from the heat capacity for the two-state system over a broad temperature interval. The consequence may be that some three-state (or even more than three-states) systems have been misinterpreted as two-state systems when the conclusion is drawn solely upon the van't Hoff enthalpy. These findings are important not only for proteins, but also for the interpretation of thermodynamical systems in general.

Protein production can be regulated at the translation stage through modulation of mRNA activity and degradation. In the unfolded protein response in S. cerevisiae it works by regulating the conversion rate from a reservoir of passive mRNA to an active short-lived mRNA that is open for translation. We develop a mathematical model for translation regulation, and elucidate its properties in perspective of the size and timing of the unfolded protein response. Optimal response is obtained when active mRNA has high decay rate compared to both the conversion rate and the decay rate of passive mRNA. In that case the translation regulation can provide the observed pulse of chaperones that fast restore protein folding conditions in the endoplasmic reticulum. Finally, we discuss translation control in relation to other known mechanisms for stress responses. Feedback on the translation level is found to be superior to transcription when conditions necessitate a fast shift in protein concentration while retaining a small cost in terms of protein degradation.

We consider highly specific protein–protein interactions in proteomes of simple model proteins. We are inspired by the work of Zarrinpar et al (2003 Nature426 676). They took a binding domain in a signalling pathway in yeast and replaced it with domains of the same class but from different organisms. They found that the probability of a protein binding to a protein from the proteome of a different organism is rather high, around one half. We calculate the probability of a model protein from one proteome binding to the protein of a different proteome. These proteomes are obtained by sampling the space of functional proteomes uniformly. In agreement with Zarrinpar et al we find that the probability of a protein binding a protein from another proteome is rather high, of order one tenth. Our results, together with those of Zarrinpar et al, suggest that designing, say, a peptide to block or reconstitute a single signalling pathway, without affecting any other pathways, requires knowledge of all the partners of the class of binding domains the peptide is designed to mimic. This knowledge is required to use negative design to explicitly design out interactions of the peptide with proteins other than its target. We also found that patches that are required to bind with high specificity evolve more slowly than those that are required only to not bind to any other patch. This is consistent with some analysis of sequence data for proteins engaged in highly specific interactions.

Cell contact, movement and directionality are important factors in biological development (morphogenesis), and myxobacteria are a model system for studying cell–cell interaction and cell organization preceding differentiation. When starved, thousands of myxobacteria cells align, stream and form aggregates which later develop into round, non-motile spores. Canonically, cell aggregation has been attributed to attractive chemotaxis, a long range interaction, but there is growing evidence that myxobacteria organization depends on contact-mediated cell–cell communication. We present a discrete stochastic model based on contact-mediated signaling that suggests an explanation for the initialization of early aggregates, aggregation dynamics and final aggregate distribution. Our model qualitatively reproduces the unique structures of myxobacteria aggregates and detailed stages which occur during myxobacteria aggregation: first, aggregates initialize in random positions and cells join aggregates by random walk; second, cells redistribute by moving within transient streams connecting aggregates. Streams play a critical role in final aggregate size distribution by redistributing cells among fewer, larger aggregates. The mechanism by which streams redistribute cells depends on aggregate sizes and is enhanced by noise. Our model predicts that with increased internal noise, more streams would form and streams would last longer. Simulation results suggest a series of new experiments.

The inherent complexity of cellular signaling networks and their importance to a wide range of cellular functions necessitates the development of modeling methods that can be applied toward making predictions and highlighting the appropriate experiments to test our understanding of how these systems are designed and function. We use methods of statistical mechanics to extract useful predictions for complex cellular signaling networks. A key difficulty with signaling models is that, while significant effort is being made to experimentally measure the rate constants for individual steps in these networks, many of the parameters required to describe their behavior remain unknown or at best represent estimates. To establish the usefulness of our approach, we have applied our methods toward modeling the nerve growth factor (NGF)-induced differentiation of neuronal cells. In particular, we study the actions of NGF and mitogenic epidermal growth factor (EGF) in rat pheochromocytoma (PC12) cells. Through a network of intermediate signaling proteins, each of these growth factors stimulates extracellular regulated kinase (Erk) phosphorylation with distinct dynamical profiles. Using our modeling approach, we are able to predict the influence of specific signaling modules in determining the integrated cellular response to the two growth factors. Our methods also raise some interesting insights into the design and possible evolution of cellular systems, highlighting an inherent property of these systems that we call 'sloppiness.'