Multi-scale genetic dynamic modelling I : an algorithm to compute generators

Abstract

We present a new approach or framework to model dynamic regulatory genetic activity. The framework is using a multi-scale analysis based upon generic assumptions on the relative time scales attached to the different transitions of molecular states defining the genetic system. At micro-level such systems are regulated by the interaction of two kinds of molecular players: macro-molecules like DNA or polymerases, and smaller molecules acting as transcription factors. The proposed genetic model then represents the larger less abundant molecules with a finite discrete state space, for example describing different conformations of these molecules. This is in contrast to the representations of the transcription factors which are-like in classical reaction kinetics-represented by their particle number only. We illustrate the method by considering the genetic activity associated to certain configurations of interacting genes that are fundamental to modelling (synthetic) genetic clocks. A largely unknown question is how different molecular details incorporated via this more realistic modelling approach lead to different macroscopic regulatory genetic models which dynamical behaviour might-in general-be different for different model choices. The theory will be applied to a real synthetic clock in a second accompanying article (Kirkilioniset al., Theory Biosci, 2011).

Multi-scale genetic dynamic modelling II: application to synthetic biology

“…6) are based on the principles laid out in Kirkilionis et al (2011). The discrete state spaces S i for the models, i ¼ 1; .…”

“…Compare these definitions with the example model in Kirkilionis et al (2011). With the help of the algorithm described in Kirkilionis et al (2011) we can construct the infinitesimal generators K i , i = 1, 2, 3, 4, 5 for the models I, II, III, IV and V (see Table 1). The corresponding transition graphs are depicted in Fig.…”

“…(2011) 130:183-201 197 numbers/concentration, and y to denote LacI molecule numbers/concentration. As suggested by our framework described in Kirkilionis et al (2011), we now need to make the step to derive the deterministic average dynamics based on the Markov chain derivations of the previous sections.…”

Multiscale Analysis in Many-Body Systems

“…This paper is intended as a general introduction to an approach aimed at studying systems (like a cell) whose processes take place at very different spatial, time and size scales (see [19][20][21]) involving different type of molecular "players". Some relevant applications of this approach can be found in [8][9][10]. The author would like to thank one of the referee for drawing his attention to [2,4,13] where the multiscale analysis is addressed from a more probabilistic point of view.…”

“…are of mass-action type, the average dynamics is no longer of that type and usually presents non-linear terms able, for instance, to describe those dynamical features usually termed generically as "nonlinear cooperativity". Novel applications of this approach to the study of synthetic genetic clock can be found in [9,10]; furthermore, the construction of the average dynamics has many features that appears in the modelling of molecular motors, see [1].…”