The mathematical model describing the stationary natural pH-gradient arising under the action of an electric field in an aqueous solution of ampholytes (amino acids) is constructed and investigated. The model is a part of a more general model of the isoelectrofocusing process. Investigation is based on the approximation of a weak solution by the piecewise continuous non-smooth functions. The method can be used for solving classes of problems for ODEs with a small parameter at higher derivatives and the turning points.

The urea-urease clock reaction is a pH switch from acid to basic that can turn into a pH oscillator if it occurs inside a suitable open reactor. We study the confinement of the reaction to lipid vesicles, which permit the exchange with an external reservoir by differential transport, enabling the recovery of the pH level and yielding a constant supply of urea molecules.For microscopically small vesicles, the discreteness of the number of molecules requires a stochastic treatment of the reaction dynamics. Our analysis shows that intrinsic noise induces a significant statistical variation of the oscillation period, which increases as the vesicles become smaller. The mean period, however, is found to be remarkably robust for vesicle sizes down to approximately 200 nm. The observed oscillations are explained as a canard-like limit cycle that differs from the wide class of conventional feedback oscillators.

An association behavior of uranyl ions in aqueous solutions is explored. For this purpose a set of all-atom molecular dynamics simulations is performed. During the simulation, the fractions of uranyl ions involved in dimer and trimer formations were monitored. To accompany the fraction statistics one also collected distributions characterizing average times of the dimer and trimer associates. Two factors effecting the uranyl association were considered: temperature and pH. As one can expect, an increase of the temperature decreases an uranyl capability of forming the associates, thus lowering bound fractions/times and vice versa. The effect of pH was modeled by adding H^+ or OH^- ions to a "neutral" solution. The addition of hydroxide ions OH^- favors the formation of the associates, thus increasing bound times and fractions. The extra H^+ ions in a solution produce an opposite effect, thus lowering the uranyl association capability. We also made a structural analysis for all the observed associates to reveal the mutual orientation of the uranyl ions.

Redox processes are important in chemistry, with applications in biomedicine, chemical analysis, among others. As many redox experiments are also performed at a fixed value of pH, having an efficient computational method to support experimental measures at both constant redox potential and pH is very important. Such computational techniques have the potential to validate experimental observations performed under these conditions and to provide additional information unachievable experimentally such as an atomic level description of macroscopic measures. We present the implementation of discrete redox and protonation states methods for constant redox potential Molecular Dynamics (CEMD), for coupled constant pH and constant redox potential MD (C(pH,E)MD), and for Replica Exchange MD along the redox potential dimension (E-REMD) on the AMBER software package. Validation results are presented for a small system that contains a single heme group: N-acetylmicroperoxidase-8 (NAcMP8) axially connected to a histidine peptide. The methods implemented allow one to make standard redox potential (Eo) predictions with the same easiness and accuracy as pKa predictions using the constant pH molecular dynamics and pH-REMD methods currently available on AMBER. In our simulations, we can correctly describe, in agreement also with theoretical predictions, the following behaviors: when a redox-active group is reduced, the pKa of a near pH-active group increases because it becomes easier for a proton to be attached; equivalently, when a pH-active group is protonated, the Eo of an adjacent redox-active group rises. Furthermore, our results also show that E-REMD is able to achieve faster statistical convergence than CEMD or C(pH,E)MD. Moreover, computational benchmarks using our methodologies show high-performance of GPU accelerated calculations in comparison to conventional CPU calculations.

Stochastic gene expression has been implicated in a variety of cellular processes, including cell differentiation and disease. In this issue of Cell, Weinberger et al. (2005) take an integrated computational-experimental approach to study the Tat transactivation feedback loop in HIV-1 and show that fluctuations in a key regulator, Tat, can result in a phenotypic bifurcation. This phenomenon is observed in an isogenic population where individual cells display two distinct expression states corresponding to latent and productive infection by HIV-1. These findings demonstrate the importance of stochastic gene expression in molecular "decision-making."

We present a detailed analysis, based on the Forward Flux Sampling (FFS) simulation method, of the switching dynamics and stability of two models of genetic toggle switches, consisting of two mutually-repressing genes encoding transcription factors (TFs); in one model (the exclusive switch), they mutually exclude each other's binding, while in the other model (general switch) the two transcription factors can bind simultaneously to the shared operator region. We assess the role of two pairs of reactions that influence the stability of these switches: TF-TF homodimerisation and TF-DNA association/dissociation. We factorise the flipping rate k into the product of the probability rho(q*) of finding the system at the dividing surface (separatrix) between the two stable states, and a kinetic prefactor R. In the case of the exclusive switch, the rate of TF-operator binding affects both rho(q*) and R, while the rate of TF dimerisation affects only R. In the case of the general switch both TF-operator binding and TF dimerisation affect k, R and rho(q*). To elucidate this, we analyse the transition state ensemble (TSE). For the exclusive switch, varying the rate of TF-operator binding can drastically change the pathway of switching, while changing the rate of dimerisation changes the switching rate without altering the mechanism. The switching pathways of the general switch are highly robust to changes in the rate constants of both TF-operator and TF-TF binding, even though these rate constants do affect the flipping rate; this feature is unique for non-equilibrium systems.

Efforts to catalogue the structure of metabolic networks have generated highly detailed, genome-scale atlases of biochemical reactions in the cell. Unfortunately, these atlases fall short of capturing the kinetic details of metabolic reactions, instead offering only \textit{topological} information from which to make predictions. As a result, studies frequently consider the extent to which the topological structure of a metabolic network determines its dynamic behavior, irrespective of kinetic details. Here, we study a class of metabolic networks known as non-autocatalytic metabolic cycles, and analytically prove an open conjecture regarding the stability of their steady-states. Importantly, our results are invariant to the choice of kinetic parameters, rate laws, equilibrium fluxes, and metabolite concentrations. Unexpectedly, our proof exposes an elementary but apparently open problem of locating the roots of a sum of two polynomials S = P+Q, when the roots of the summand polynomials P and Q are known. We derive two new results named the Stubborn Roots Theorems, which provide sufficient conditions under which the roots of S remain qualitatively identical to the roots of P. Our work illustrates how complementary feedback, from classical fields like dynamical systems to biology and vice versa, can expose fundamental and potentially overlooked questions.

We have developed a mathematical model of regulation of expression of the Escherichia coli lac operon, and have investigated bistability in its steady-state induction behavior in the absence of external glucose. Numerical analysis of equations describing regulation by artificial inducers revealed two natural bistability parameters that can be used to control the range of inducer concentrations over which the model exhibits bistability. By tuning these bistability parameters, we found a family of biophysically reasonable systems that are consistent with an experimentally determined bistable region for induction by thio-methylgalactoside (Ozbudak et al. Nature 427:737, 2004). The model predicts that bistability can be abolished when passive transport or permease export becomes sufficiently large; the former case is especially relevant to induction by isopropyl-beta, D-thiogalactopyranoside. To model regulation by lactose, we developed similar equations in which allolactose, a metabolic intermediate in lactose metabolism and a natural inducer of lac, is the inducer. For biophysically reasonable parameter values, these equations yield no bistability in response to induction by lactose; however, systems with an unphysically small permease-dependent export effect can exhibit small amounts of bistability for limited ranges of parameter values. These results cast doubt on the relevance of bistability in the lac operon within the natural context of E. coli, and help shed light on the controversy among existing theoretical studies that address this issue. The results also suggest an experimental approach to address the relevance of bistability in the lac operon within the natural context of E. coli.

Optimal trade execution is an important problem faced by essentially all traders. Much research into optimal execution uses stringent model assumptions and applies continuous time stochastic control to solve them. Here, we instead take a model free approach and develop a variation of Deep Q-Learning to estimate the optimal actions of a trader. The model is a fully connected Neural Network trained using Experience Replay and Double DQN with input features given by the current state of the limit order book, other trading signals, and available execution actions, while the output is the Q-value function estimating the future rewards under an arbitrary action. We apply our model to nine different stocks and find that it outperforms the standard benchmark approach on most stocks using the measures of (i) mean and median out-performance, (ii) probability of out-performance, and (iii) gain-loss ratios.

The Tsallis $q$-Gaussian distribution is a powerful generalization of the standard Gaussian distribution and is commonly used in various fields, including non-extensive statistical mechanics, financial markets and image processing. It belongs to the $q$-distribution family, which is characterized by a non-additive entropy. Due to their versatility and practicality, $q$-Gaussians are a natural choice for modeling input quantities in measurement models. This paper presents the characteristic function of a linear combination of independent $q$-Gaussian random variables and proposes a numerical method for its inversion. The proposed technique makes it possible to determine the exact probability distribution of the output quantity in linear measurement models, with the input quantities modeled as independent $q$-Gaussian random variables. It provides an alternative computational procedure to the Monte Carlo method for uncertainty analysis through the propagation of distributions.