Modelling molecular-level processes using parametric extension of L-systems

L-systems are a useful framework for modelling the development of growing linear and branching structures of organisms, from herbaceous plants to trees to entire plant ecosystems. They can be used to model molecular-level processes such as the regulation of growth and differentiation by genetic regulatory networks, source–sink interactions, and metabolic regulation. Until now, the dynamics of these processes has been expressed using differential equations, implying continuously-valued concentrations of the substances involved. That assumption is not satisfied in many biological processes when the numbers of molecules are relatively low.

In their new paper published in in silico Plants, Cieslak and Prusinkiewicz of the Biological Modeling and Visualization research group in the Department of Computer Science at the University of Calgary, propose and test the integrated L-systems and the Gillespie Stochastic Simulation Algorithm to simulate stochastic processes in fixed and developing linear structures.

According to Senior Research Associate Mikolaj Cieslak, “Gillespie’s method is a well-known simulation technique for discrete biochemical kinetics and is accurate even when the number of molecules is small. Its integration with L-systems offers a convenient framework for studying the effect of noise on developmental processes in nature.”

The authors illustrate the strength of Gillespie L-systems using examples of morphogenetic processes that include reaction-diffusion and auxin driven patterning. “In each case, we were able to highlight the impact that the number of molecules has on the characteristics of the solution,” says Professor Przemyslaw Prusinkiewicz. The authors also show that, as expected, the stochastic solutions converge to their continuous counterparts as the number of molecules increases.

The presented method and software can be used to simulate molecular and higher-level spatially explicit stochastic processes in static and developing structures, and study their behavior in the presence of stochastic perturbations.

The software to run the models is available on the lab’s website: http://www.algorithmicbotany.org

The models are available upon request from the authors.