A delay stochastic process with applications in molecular biology.
Molecular processes of cell differentiation often involve reactions with delays. We develop a mathematical model that provides a basis for a rigorous theoretical analysis of these processes as well as for direct simulation. A discrete, stochastic approach is adopted because several molecules appear in small numbers only. Our model is a non-Markovian stochastic process. The main theoretical results include a constructive proof of the existence of the process and a derivation of the rates for initiation and completion of reactions with delays. These results guarantee that the stochastic process is a consistent and realistic description of the molecular system. They also serve as a theoretical justification of recent work on delay stochastic simulation. We apply our model to an important process in developmental biology, the formation of somites in the vertebrate embryo. Simulation of the molecular oscillator controlling this process reveals major differences between stochastic and deterministic models.
Schlicht
Robert
R
Institute of Biomathematics and Biometry, Ingolstädter Landstr. Neuherberg, Germany. robert.schlicht@helmholtz-muenchen.de
Winkler
Gerhard
G
D000465:Algorithms; D000818:Animals; D051792:Basic Helix-Loop-Helix Proteins / Q000235:genetics / Q000502:physiology; D019295:Computational Biology / Q000379:methods*; D003198:Computer Simulation; D018507:Gene Expression Regulation, Developmental; D047908:Intracellular Signaling Peptides and Proteins; D008565:Membrane Proteins / Q000235:genetics / Q000502:physiology; D008648:Mesoderm / Q000196:embryology / Q000378:metabolism; D008957:Models, Genetic; D051880:Receptors, Notch / Q000235:genetics / Q000502:physiology; D019170:Somites / Q000196:embryology / Q000378:metabolism*; D013269:Stochastic Processes; D014157:Transcription Factors / Q000235:genetics / Q000502:physiology; D015027:Zebrafish; D029961:Zebrafish Proteins / Q000235:genetics / Q000502:physiology
D051792:Basic Helix-Loop-Helix Proteins; D047908:Intracellular Signaling Peptides and Proteins; D008565:Membrane Proteins; D051880:Receptors, Notch; D014157:Transcription Factors; D029961:Zebrafish Proteins; C460099:HER7 protein, zebrafish; C064045:delta protein; C491619:her1 protein, zebrafish