Macroscopic theory

At the roughest level of description, we are using mathematical models of mass reaction kinetics to study the dynamics of protocellular systems.

Reaction kinetics coupling of gene and container replication

Primary investigator: Steen Rasmussen

Template-directed replication is known to obey a parabolic growth law due to product inhibition (Sievers & Von Kiedrowski, 1994 Nature 369, 221; Lee et al., 1996 Nature 382, 525; Varga & Szathmary, 1997, Bull. Math. Biol59, 1145). We investigate a template-directed replication with a coupled template catalyzed lipid aggregate production as a model of a minimal protocell and show analytically that the autocatalytic template–container feedback ensures balanced exponential replication kinetics; both the genes and the container grow exponentially with the same exponent. The parabolic gene replication does not limit the protocellular growth, and a detailed stoichiometric control of the individual protocell components is not necessary to ensure a balanced gene–container growth as conjectured by various authors (Ganti, 2004, Chemoton theory. Our analysis also suggests that the exponential growth of most modern biological systems emerges from the inherent spatial quality of the container replication process as we show analytically how the internal gene and metabolic kinetics determine the cell population's generation time and not the growth law (Burdett & Kirkwood, 1983, J. Theor. Biol. 103, 11-20; Novak et al., 1998, Biophys. Chem. 72, 185-200; Tyson et al., 2003, Curr. Opin. Cell Biol. 15, 221-231). Previous extensive replication reaction kinetic studies have mainly focused on template replication and have not included a coupling to metabolic container dynamics (Stadler et al., 2000, Bull. Math. Biol. 62, 1061-1086; Stadler & Stadler, 2003, Adv. Comp. Syst. 6, 47). The reported results extend these investigations. Finally, the coordinated exponential gene-container growth law stemming from catalysis is an encouraging circumstance for the many experimental groups currently engaged in assembling self-replicating minimal artificial cells (Szostak et al., 2001, Nature 409, 387-390; Pohorille & Deamer, 2002, Trends Biotech. 20 123-128; Rasmussen et al., 2004, Science 963-965; Szathmary, 2005, Nature 433, 469-470; Luisi et al., 2006, Naturwissenschaften 93, 1-13).

The main results from this work from this work are summarized here: The template catalyses the lipid production while the lipid aggregate makes possible (catalyses) the template replication. Because the local template concentration is kept approximately constant due to the aggregate growth the template replicates exponentially. The generation (doubling) time is given by , where all parameters in principle are experimentally observable quantities (for details see Roucheleau et al., 2007).

Related publications:

Rochelau, T.; Rasmussen, S.; Nielsen, P.; Jacobi, M.; Ziock, H. (2007). Emergence of protocellular growth laws. In: Phil. Trans. R. Soc B., 362(1486): 1841-1845.

Metabolic photo-fragmentation reaction kinetics

Primary investigators: Steen Rasmussen and Goran Goranovic

A key requirement of an autonomous self-replicating molecular machine, a protocell, is the ability to digest resources and turn them into building blocks. Thus a protocell needs a set of metabolic processes fueled by external free energy in the form of available chemical redox potential or light. We introduce and investigate a minimal photo-driven metabolic system, which is based on photofragmentation of resource molecules catalyzed by genetic molecules. We represent and analyze the full metabolic set of reaction kinetic equations and, through a set of approximations, simplify the reaction kinetics such that analytical expressions can be obtained for the building block production. The analytical approximations are compared to the full equation set and to corresponding experimental results to the extent they are available. It should be noted, however, that the proposed metabolic system has not been experimentally implemented, so this investigation is conducted to obtain a deeper understanding of its dynamics and perhaps to anticipate its limitations. We demonstrate that this type of minimal photo-driven metabolic scheme is typically rate limited by the front-end photoexcitation process while its yield is determined by the genetic catalysis. We further predict how gene catalyzed metabolic reactions can only undergo evolutionary selection for certain combinations of the involved reaction rates due to their intricate interactions. We finally discuss how the expected range of metabolic rates likely impacts other key protocellular processes such as container growth and division as well as gene replication.

Related publications:

C. Knutson, G. Benkö, T. Rocheleau, F. Mouffouk, J. Maselko, L. Chen, A. Shreve, and S. Rasmussen, Artificial Life 14 (2008) 189-201.