arXiv subjects

Tiejun Li

Publications and source records attributed to Tiejun Li.

41 records · Page 3Linked to original sources

arXiv2016

Two-scale large deviations for chemical reaction kinetics through second quantization path integral

Motivated by the study of rare events for a typical genetic switching model in systems biology, in this paper we aim to establish the general two-scale large deviations for chemical reaction systems. We build a formal approach to explicitly obtain the large deviation rate functionals for the considered two-scale processes based upon the second-quantization path integral technique. We get three important types of large deviation results when the underlying two times scales are in three different regimes. This is realized by singular perturbation analysis to the rate functionals obtained by path integral. We find that the three regimes possess the same deterministic mean-field limit but completely different chemical Langevin approximations. The obtained results are natural extensions of the classical large volume limit for chemical reactions. We also discuss its implication on the single-molecule Michaelis-Menten kinetics. Our framework and results can be applied to understand general multi-scale systems including diffusion processes.

cond-mat.stat-mech↗

arXiv2015

Realization of Waddington's Metaphor: Potential Landscape, Quasi-potential, A-type Integral and Beyond

Motivated by the famous Waddington's epigenetic landscape metaphor in developmental biology, biophysicists and applied mathematicians made different proposals to realize this metaphor in a rationalized way. We adopt comprehensive perspectives to systematically investigate three different but closely related realizations in recent literature: namely the potential landscape theory from the steady state distribution of stochastic differential equations (SDEs), the quasi-potential from the large deviation theory, and the construction through SDE decomposition and A-type integral.The connections among these theories are established in this paper. We demonstrate that the quasi-potential is the zero noise limit of the potential landscape. We also show that the potential function in the third proposal coincides with the quasi-potential. The most probable transition path by minimizing the Onsager-Machlup or Freidlin-Wentzell action functional is discussed as well. Furthermore, we compare the difference between local and global quasi-potential through the exchange of limit order for time and noise amplitude. As a consequence of such explorations, we arrive at the existence result for the SDE decomposition while deny its uniqueness in general cases. It is also clarified that the A-type integral is more appropriate to be applied to the decomposed SDEs rather than the original one. Our results contribute to a better understanding of existing landscape theories for biological systems.

q-bio.MN↗

arXiv2014

Finding Transition Pathways on Manifolds

We consider noise-induced transition paths in randomly perturbed dynami- cal systems on a smooth manifold. The classical Freidlin-Wentzell large devia- tion theory in Euclidean spaces is generalized and new forms of action functionals are derived in the spaces of functions and the space of curves to accommodate the intrinsic constraints associated with the manifold. Numerical meth- ods are proposed to compute the minimum action paths for the systems with constraints. The examples of conformational transition paths for a single and double rod molecules arising in polymer science are numerically investigated.

math-ph↗

arXiv2012

Transition Path, Quasi-potential Energy Landscape and Stability of Genetic Switches

One of the fundamental cellular processes governed by genetic regulatory networks in cells is the transition among different states under the intrinsic and extrinsic noise. Based on a two-state genetic switching model with positive feedback, we develop a framework to understand the metastability in gene expressions. This framework is comprised of identifying the transition path, reconstructing the global quasi-potential energy landscape, analyzing the uphill and downhill transition paths, etc. It is successfully utilized to investigate the stability of genetic switching models and fluctuation properties in different regimes of gene expression with positive feedback. The quasi-potential energy landscape, which is the rationalized version of Waddington potential, provides a quantitative tool to understand the metastability in more general biological processes with intrinsic noise.

q-bio.MN↗

arXiv2006

On the Poisson Approximation to Photon Distribution for Faint Lasers

It is proved, that for a certain kind of input distribution, the strongly binomially attenuated photon number distribution can well be approximated by a Poisson distribution. This explains why we can adopt poissonian distribution as the photon number statistics for faint lasers. The error of such an approximation is quantitatively estimated. Numerical tests are carried out, which coincide with our theoretical estimations. This work lays a sound mathematical foundation for the well-known intuitive idea which has been widely used in quantum cryptography.

math-ph↗