A Dynamical Low-Rank Approach to the Chemical Master Equation

Stochastic reaction kinetics have increasingly been used to study cellular systems, with applications ranging from viral replication to gene regulatory networks and to signaling pathways. The underlying evolution equation, known as the chemical master equation (CME), can rarely be solved with traditional methods due to the huge number of degrees of freedom. We present a new approach to directly solve the CME by a dynamical low-rank approximation based on the Dirac–Frenkel–McLachlan variational principle. The new approach has the capability to substantially reduce the number of degrees of freedom, and to turn the CME into a computationally tractable problem. We illustrate the accuracy and efficiency of our methods in application to two examples of biological interest.     

Direct Solution of the Chemical Master Equation Using Quantized Tensor Trains

The Chemical Master Equation (CME) is a cornerstone of stochastic analysis and simulation of models of biochemical reaction networks. Yet direct solutions of the CME have remained elusive. Although several approaches overcome the infinite dimensional nature of the CME through projections or other means, a common feature of proposed approaches is their susceptibility to the curse of dimensionality, i.e. the exponential growth in memory and computational requirements in the number of problem dimensions. We present a novel approach that has the potential to “lift” this curse of dimensionality. The approach is based on the use of the recently proposed Quantized Tensor Train (QTT) formatted numerical linear algebra for the low parametric, numerical representation of tensors. The QTT decomposition admits both, algorithms for basic tensor arithmetics with complexity scaling linearly in the dimension (number of species) and sub-linearly in the mode size (maximum copy number), and a numerical tensor rounding procedure which is stable and quasi-optimal. We show how the CME can be represented in QTT format, then use the exponentially-converging -discontinuous Galerkin discretization in time to reduce the CME evolution problem to a set of QTT-structured linear equations to be solved at each time step using an algorithm based on Density Matrix Renormalization Group (DMRG) methods from quantum chemistry. Our method automatically adapts the “basis” of the solution at every time step guaranteeing that it is large enough to capture the dynamics of interest but no larger than necessary, as this would increase the computational complexity. Our approach is demonstrated by applying it to three different examples from systems biology: independent birth-death process, an example of enzymatic futile cycle, and a stochastic switch model. The numerical results on these examples demonstrate that the proposed QTT method achieves dramatic speedups and several orders of magnitude storage savings over direct approaches.     

A Hybrid of the Chemical Master Equation and the Gillespie Algorithm for Efficient Stochastic Simulations of Sub-Networks

Modeling stochastic behavior of chemical reaction networks is an important endeavor in many aspects of chemistry and systems biology. The chemical master equation (CME) and the Gillespie algorithm (GA) are the two most fundamental approaches to such modeling; however, each of them has its own limitations: the GA may require long computing times, while the CME may demand unrealistic memory storage capacity. We propose a method that combines the CME and the GA that allows one to simulate stochastically a part of a reaction network. First, a reaction network is divided into two parts. The first part is simulated via the GA, while the solution of the CME for the second part is fed into the GA in order to update its propensities. The advantage of this method is that it avoids the need to solve the CME or stochastically simulate the entire network, which makes it highly efficient. One of its drawbacks, however, is that most of the information about the second part of the network is lost in the process. Therefore, this method is most useful when only partial information about a reaction network is needed. We tested this method against the GA on two systems of interest in biology - the gene switch and the Griffith model of a genetic oscillator—and have shown it to be highly accurate. Comparing this method to four different stochastic algorithms revealed it to be at least an order of magnitude faster than the fastest among them.     

On a Nonlinear Master Equation and the Haken-Kelso-Bunz Model

A nonlinear master equation (NLME) is proposed basedon general information measures.Classical and cut-off solutions of the NLME are considered.In the former case, the NLME exhibits uniquely defined stationary distributions. In the latter case, there are multiple stationary distributions.In particular, for classical solutions, it is shown that transient solutions converge to stationary distributions that maximize information measures (H-theorem). Cut-off distributions arestudied numerically for the Haken-Kelso-Bunz model. The Haken-Kelso-Bunz modelis known to describe multistable human motor control systems. It is shownthat a stochastic Haken-Kelso-Bunz model based on a NLME can exhibit multiplestationary cut-off distributions.In doing so, we illustrate that multistability in stochastic biological systems can beestablished by means of cut-off distributions.     

一阶非线性多时滞微分方程的振动性

考虑一阶非线性多时滞微分方程x′（t）=f（t,x（t-r₁）,x（t-r₂）,…,x（t-r_n））,利用其线性近似方程x′（t）=∑i=1nD_i+1f（t,0,…,0）x（t-r_i）的振动性,给出了方程解振动的一个充分条件,所得结果推广了文献[5]的相关结果.     

Chemical Reduction of Phosphate on the Primitive Earth

If phosphorus played a role in the origin of life, some means of concentrating micromolar levels of phosphate (derived from the calcium phosphate mineral apatite), must first have been available. Here we show that simulated (mini)lightning discharges in model prebiotic atmospheres, including only minimally reducing ones, reduce orthophosphates, including apatite, to produce substantial yields of phosphite. Electrical discharges associated with volcanic eruptions could have provided a particularly suitable environment for this process. Production of relatively soluble and reactive phosphite salts could have supplied a pathway by which the first phosphorus atoms were incorporated into (pre)biological systems.     

Universal Optimality of Nearest Neighbour Balanced Block Designs using Second Order Correlated Models

Block designs for observations correlated in one dimension are investigated. Santharam and Ponnusamy (1995) investigated the universal optimality of Nearest Neighbour balanced block designs (NNBD) using first order correlated models (AR(1), MA(1) and ARMA(1,1)). In this article we have investigated the universal optimality of NNBD using second order correlated models (ar(2), and MA (2)).     

Studies on Order in Prebiological Systems at the Laboratory of Chemical Evolution

The basic tenet of investigations in the Laboratory of Chemical Evolution (LCE) under Cyril Ponnamperuma was that biology is a recapitulation of prebiology, and that protobiology is an outcome of simple molecular interactions, engendered by the physics and chemistry of the molecules themselves. Studies were undertaken to continue research into understanding the determining physical and chemical parameters of molecular interactions leading to increasing complexity in pre- and proto-biological systems. Among other, related work, research was performed on the origin of the genetic code, the origin of order in prebiotic polymers, and related studies on the origins of optical activity in biological macromolecules. Highlights of some these studies are presented here.     

Thiosulfate Reduction, an Important Physiological Feature Shared by Members of the Order Thermotogales

Several members of the order Thermotogales in the domain Bacteria, viz., Thermotoga neapolitana, Thermotoga maritima, Thermosipho africanus, Fervidobacterium islandicum, and Thermotoga strain SEBR 2665, an isolate from an oil well, reduced thiosulfate to sulfide. This reductive process enhanced cellular yields and growth rates of all the members but was more significant with the two hyperthermophiles T. neapolitana and T. maritima. This is the first report of such an occurrence in this group of thermophilic and hyperthermophilic anaerobic bacteria. The results suggest that thiosulfate reduction is important in the geochemical cycling of sulfur in anaerobic thermal environments such as the slightly acidic and neutral-pH volcanic hot springs and oil reservoirs.     

一类三阶非线性递归差分方程的全局渐近稳定性与振动性(英文)

研究了一类三阶方程x_(n+1)=x_nx_(n-2)+a/x_n+x_(n-2)+b,n=0,1,…的解的振动性和正解的全局渐近稳定性,证明了正平衡解的全局渐近稳定,非平衡解具有规律的振动性.     

On the Construction of Balanced and Partially Balanced Ternary Design

A method of constructing balanced and partially balanced ternary designs from balanced and partially balanced incomplete block designs, respectively, and two methods of constructing partially balanced ternary designs from association schemes are obtained. Two new and efficient balanced ternary designs having K < V and R ≦ 20 are obtained by the first method.     

Reduction of airway anion secretion via CFTR in sphingomyelin pathway

The present study concerns the involvement of the ceramide produced through sphingomyelinase (SMase)-mediated catalysis in airway anion secretion of Calu-3 cells. Short-circuit current (Isc) measurement revealed that isoproterenol (ISO, 0.1 microM)-induced anion secretion was prevented by pretreatment with SMase (0.3 U/ml, for 30 min) from the basolateral but not the apical side, although basal and 1-ethyl-2-benzimidazolinone (1-EBIO, a Ca2+-activated K+ channel opener)-induced Isc were unaffected. The effects of SMase were reproduced in responses to forskolin (20 microM) or 8-bromo-cAMP (2 mM). C2-ceramide, a cell-permeable analog, also repressed the 8-bromo-cAMP-induced responses. Nystatin permeabilization studies confirmed that the SMase- and C2-ceramide-induced repressions were due to hindrance of augmentation of cystic fibrosis transmembrane conductance regulator (CFTR)-mediated conductance across the apical membrane. Further, SMase failed to influence K+ conductance across the basolateral membrane. These results suggest that the ceramide originating from basolateral sphingomyelin acts on activated CFTR from the cytosolic side, hindering anion secretion.     

Increased Penicillin Production in Penicillium chrysogenum Production Strains via Balanced Overexpression of Isopenicillin N Acyltransferase

Intense classical strain improvement has yielded industrial Penicillium chrysogenum strains that produce high titers of penicillin. These strains contain multiple copies of the penicillin biosynthesis cluster encoding the three key enzymes: δ-(l-α-aminoadipyl)-l-cysteinyl-d-valine synthetase (ACVS), isopenicillin N synthase (IPNS), and isopenicillin N acyltransferase (IAT). The phenylacetic acid coenzyme A (CoA) ligase (PCL) gene encoding the enzyme responsible for the activation of the side chain precursor phenylacetic acid is localized elsewhere in the genome in a single copy. Since the protein level of IAT already saturates at low cluster copy numbers, IAT might catalyze a limiting step in high-yielding strains. Here, we show that penicillin production in high-yielding strains can be further improved by the overexpression of IAT while at very high levels of IAT the precursor 6-aminopenicillic acid (6-APA) accumulates. Overproduction of PCL only marginally stimulates penicillin production. These data demonstrate that in high-yielding strains IAT is the limiting factor and that this limitation can be alleviated by a balanced overproduction of this enzyme.     

Impact Characterization in the Tool for the Reduction and Assessment of Chemical and Other Environmental Impacts

The tool for the reduction and assessment of chemical and other environmental impacts (TRACI) is a set of life-cycle impact assessment (LCIA) characterization methods that has been developed by a series of U.S. Environmental Protection Agency research projects. TRACI facilitates the characterization of stressors that may have potential effects, including ozone depletion, global warming, acidification, eutrophication, tropospheric ozone (smog) formation, eco-toxicity, human particulate effects, human carcinogenic effects, human non-carcinogenic effects, fossil fuel depletion, and land-use effects. This article describes the methodologies developed to address acidification, eutrophication, and smog. Each of these methods offers the ability to take account of differences in expected strength of impact as a function of pollution release location within North America. Specifically, the methods employ regionalized fate and transport modeling. The resulting factors differ regionally by up to more than an order of magnitude.     

Identification of Protein Transport Complexes in the Chloroplastic Envelope Membranes via Chemical Cross-Linking

Transport of cytoplasmically synthesized proteins into chloroplasts uses an import machinery present in the envelope membranes. To identify the components of this machinery and to begin to examine how these components interact during transport, chemical cross-linking was performed on intact chloroplasts containing precursor proteins trapped at a particular stage of transport by ATP limitation. Large crosslinked complexes were observed using three different reversible homobifunctional cross-linkers. Three outer envelope membrane proteins (OEP86, OEP75, and OEP34) and one inner envelope membrane protein (IEP110), previously reported to be involved in protein import, were identified as components of these complexes. In addition to these membrane proteins, a stromal member of the hsp100 family, ClpC, was also present in the complexes. We propose that ClpC functions as a molecular chaperone, cooperating with other components to accomplish the transport of precursor proteins into chloroplasts. We also propose that each envelope membrane contains distinct translocation complexes and that a portion of these interact to form contact sites even in the absence of precursor proteins.     

二阶Volterra型积分微分差分方程非线性边值问题的存在性与唯一性

本文利用微分不等式技巧研究了某一类Volterra型二阶积分微分差分方程非线性边值问题,在上下解存在的条件下,得到了解的存在性和唯一性定理.结果表明:这种技巧为其它边值问题的研究提出了崭新的思路.