Totally Asymmetric Simple Exclusion Processes with Finite Resources

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Virginia Tech

In many situations in the world, the amount of resources available for use is limited. This statement is especially true in the cells of living organisms. During the translation process in protein synthesis, ribosomes move along the mRNA strand constructing proteins based on the sequence of codons that form a gene. The totally asymmetric simple exclusion process (TASEP) models well the translation process. However, these genes are constantly competing for ribosomes and other resources in the cell. To see how finite resources and competition affects such a system, we must construct a simple model to account for the limited resources.

We consider coupling multiple TASEPs to a finite reservoir of particles where the entry rate of particles into the TASEPs depends on the number of particles left in the reservoir. Starting with a single TASEP connected to the reservoir, we study the system using both Monte Carlo simulations and theoretical approaches. We explore how the average overall density, density profile, and current change as a function of the number of particles initially in the reservoir for various parameters. New features arise not seen in the ordinary TASEP model, even for a single TASEP connected to the pool of particles. These features include a localized shock in the density profile. To explain what is seen in the simulations, we use an appropriately generalized version of a domain wall theory.

The dynamics of the TASEPs with finite resources are also studied through the power spectra associated with the total particle occupancy of each TASEP and the reservoir. Again, we find new phenomena not seen in the power spectrum of the ordinary TASEP. For a single constrained TASEP, we find a suppression at low frequencies when compared to the power spectrum of the ordinary TASEP. The severity of this suppression is found to depend on how the entry rate changes with respect to the number of particles in the pool. For two TASEPs of different lengths, we find an enhancement of the power spectrum of the smaller TASEP when compared to the ordinary TASEP's power spectrum. We explain these findings using a linearized Langevin equation.

Finally, we model competition between ten genes found in Escherichia coli using a modified version of the TASEP. This modified version includes extended objects and inhomogeneous internal hopping rates. We use the insight gained from the previous studies of finite resources and competition as well as other studies to gain some insight into how competition affects protein production.

power spectra, domain wall, protein synthesis, open-boundary, TASEP