A data-driven method to learn a jump diffusion process from aggregate biological gene expression data.

Dynamic models of gene expression are urgently required. In this paper, we describe the time evolution of gene expression by learning a jump diffusion process to model the biological process directly. Our algorithm needs aggregate gene expression data as input and outputs the parameters of the jump diffusion process. The learned jump diffusion process can predict population distributions of gene expression at any developmental stage, obtain long-time trajectories for individual cells, and offer a novel approach to computing RNA velocity. Moreover, it studies biological systems from a stochastic dynamic perspective. Gene expression data at a time point, which is a snapshot of a cellular process, is treated as an empirical marginal distribution of a stochastic process. The Wasserstein distance between the empirical distribution and predicted distribution by the jump diffusion process is minimized to learn the dynamics. For the learned jump diffusion process, its trajectories correspond to the development process of cells, the stochasticity determines the heterogeneity of cells, its instantaneous rate of state change can be taken as "RNA velocity", and the changes in scales and orientations of clusters can be noticed too. We demonstrate that our method can recover the underlying nonlinear dynamics better compared to previous parametric models and the diffusion processes driven by Brownian motion for both synthetic and real world datasets. Our method is also robust to perturbations of data because the computation involves only population expectations.

Steady-state joint distribution for first-order stochastic reaction kinetics.

While the analytical solution for the marginal distribution of a stochastic chemical reaction network has been extensively studied, its joint distribution, i.e., the solution of a high-dimensional chemical master equation, has received much less attention. Here we develop an alternative method of computing the exact joint distributions of a wide class of first-order stochastic reaction systems in steady-state conditions. The effectiveness of our method is validated by applying it to four gene expression models of biological significance, including models with 2A peptides, nascent mRNA, gene regulation, translational bursting, and alternative splicing.

Phenotypic equilibrium as probabilistic convergence in multi-phenotype cell population dynamics.

We consider the cell population dynamics with n different phenotypes. Both the Markovian branching process model (stochastic model) and the ordinary differential equation (ODE) system model (deterministic model) are presented, and exploited to investigate the dynamics of the phenotypic proportions. We will prove that in both models, these proportions will tend to constants regardless of initial population states ("phenotypic equilibrium") under weak conditions, which explains the experimental phenomenon in Gupta et al.'s paper. We also prove that Gupta et al.'s explanation is the ODE model under a special assumption. As an application, we will give sufficient and necessary conditions under which the proportion of one phenotype tends to 0 (die out) or 1 (dominate). We also extend our results to non-Markovian cases.

[Case-control study on minimally invasive percutaneous plate osteosynthesis for the treatment of distal tibial comminuted fractures at different operation times].

OBJECTIVE: To compare clinical outcomes of minimally invasive percutaneous plate osteosynthesis (MIPPO) in treating distal tibial comminuted fractures at early and delayed stage. METHODS: From January 2006 to January 2012,66 patients with distal tibial comminuted fractures were treated by MIPPO. All patients were divided into primary group and delayed group according to operation time. There were 31 patients in primary group, including 18 males and 13 females aged 21 to 57 years old with an average of (39.0 +/- 17.8), treated by MIPPO at primary stage,according to Tscherne soft tissue injury, 18 cases were grade I ,12 cases were grade II and 1 case were grade III. Thirty-five patients were treated by MIPPO at delayed stage, including 16 males and 19 females aged 24 to 55 years old with an average of (39.5 +/- 15.2), according to Tscherne soft tissue injury, 6 cases were grade I, 26 cases were grade II and 3 cases were grade III. Operation time, blood loss, hospital stay, fracture healing time and complications of two groups were recorded and observed, Lowa scoring of ankle joint were used to evaluated therapeutic effects at final following and AP and lateral X-rays were used to evaluated fracture reduction and alignment. RESULTS: All patients were followed up, the time of following-up of primary group was (13.5 +/- 3.5) months, (15.2 +/- 3.8) months in delayed group, there was no significant meaning between two groups (t = 1.882, P = 0.064). There was no significant differences between two groups in operation time and blood loss (P > 0.05), but hospital stay in primary group was shorter than that of delayed group(P<0.05). There was no significant meaning between primary group (5.5 +/- 2.8) and delayed group (6.2 +/- 3.1) in fracture healing time (t = 0.958, P = 0.342); there was no significant meaning between primary group (87.6 +/- 6.8) and delayed group (89.6 +/- 5.2) in Lowa scores at final following-up (t = 1.351, P = 0.182). Two cases occurred postoperative superficial inflammatory reaction around fibular incision in primary group, 1 case occurred postoperative superficial inflammatory reaction around fibular incision and 1 case occurred delayed deep incision infection in delayed group at four months after operation. There was no significant differences in incidence of postoperative soft tissue complications between primary group (6.5%) and delayed group (5.7%) (t = 0.016, P = 0.900). CONCLUSION: For distal tibial comminuted fractures with grade I and II of Tscherne soft tissue injury, MIPPO at primary stage can not increase incidence of soft tissue complications, also can obtain the same clinical outcomes just like delayed MIPPO.

Kinetic behavior of the general modifier mechanism of Botts and Morales with non-equilibrium binding.

In this paper, we perform a complete analysis of the kinetic behavior of the general modifier mechanism of Botts and Morales in both equilibrium steady states and non-equilibrium steady states (NESS). Enlightened by the non-equilibrium theory of Markov chains, we introduce the net flux into discussion and acquire an expression of the rate of product formation in NESS, which has clear biophysical significance. Up till now, it is a general belief that being an activator or an inhibitor is an intrinsic property of the modifier. However, we reveal that this traditional point of view is based on the equilibrium assumption. A modifier may no longer be an overall activator or inhibitor when the reaction system is not in equilibrium. Based on the regulation of enzyme activity by the modifier concentration, we classify the kinetic behavior of the modifier into three categories, which are named hyperbolic behavior, bell-shaped behavior, and switching behavior, respectively. We show that the switching phenomenon, in which a modifier may convert between an activator and an inhibitor when the modifier concentration varies, occurs only in NESS. Effects of drugs on the Pgp ATPase activity, where drugs may convert from activators to inhibitors with the increase of the drug concentration, are taken as a typical example to demonstrate the occurrence of the switching phenomenon.