RouletteCyte : A Stochastic Cell Simulator

The relatively small numbers of key intracellular macromolecules suggests that deviation from macroscopic chemical kinetic behavior can be significant. For example, an initially synchronous cell culture can become asynchronous in time. Furthermore, a collection of cells (e.g. a tumor) can develop a spectrum of cell states due to mutation or other random effects during the cell cycle. RouletteCyte is a new stochastic model of transcription, translation and metabolics, focusing on the states of the many individual enzymes, ribosomes and other structures as they progress through a set of available states during polymerization to form mRNA and proteins to arrive at a stochastic genomic/proteomic/metabolic model. Our formulation is based on a Markov approach.

Since McQuarrie (1976) it has been recognized that small reacting systems behave stochastically. In the McQuarrie formulation, one develops a master equation for the probability of discrete variables that monitor the numbers of molecules of each type in the system. Thus, dynamics is described in terms of the probability of transition between these discrete states; the transition probabilities are designed to capture the macroscopic chemical kinetics of the fluctuation-free system. Equivalent to a master equation is a Monte-Carlo approach that describes the evolution of an individual system and not a whole ensemble of systems (as with the master equation approach). The weakness of this approach is that one must simulate many realizations of the fluctuating dynamics in order to obtain ensemble average properties (i.e. the behavior of a collection of cells). The strength of this method is that it can be applied to complex systems with many state variables so that the joint probability arising in the master equation approach cannot be solved on existing hardware.

Our approach follows each RNA polymerase molecule and ribosome as they pass through their transcription and translation steps. This is done via a step-by-step polymerization model, which accounts for the consumption of nucleotides and amino acids. This dynamic is coupled to the detailed metabolics of nucleotide and amino acid production using equations that follow from multiple time scale arguments and the resulting Markov process. Running RouletteCyte yields the fluctuating time course of all the ribosomes and RNA polymerases, as well as the resulting populations of mRNA and proteins. Through averaging, the results can be used to predict the dynamics of a tumor or other populations of cells. In contrast to McAdams and Arkin (1997,1998), RouletteCyte does not ignore the many fluctuating polymerization steps and allows for full coupling between the metabolome, proteome and genome. In the problem of interest here, the fluctuating dynamics of the genome/proteome/metabolome (biome) system, there are many states possible for each of a large number of degrees of freedom.

Consider the polymerization model of Weitzke and Ortoleva (2003) of prokaryote transcription and translation. If one focuses on the transcription enzyme structures and the ribosomes, these objects can be at any of a large number of positions on the DNA at each gene (for the enzyme) and on any of the mRNA (for the ribosomes). There are many enzymes and ribosomes acting simultaneously so that to solve the associated master equation one must construct a joint probability for many variables, each of which can have many possible values. For example, there are many simultaneously active ribosomes, each of which can be at any of many possible mRNA present at any time.

Besides these technical difficulties in solving the master equation for the stochastic dynamics of an ensemble of cells, there is the more interesting question as to how many individual cells can evolve in a rather similar way despite all the uncertainty in their dynamics. To address this question, one must follow a number of individuals through their fluctuating scenario and then determine the manner in which they are able to thwart uncertainty to live a rather typical (average) life cycle. Departures from this average are also of interest.