Eduardo S. Zeron

A Laplace transform algorithm for the volume of a convex polytope

We provide two algorithms for computing the volume of the convex polytope Ω : = {<i>x</i> ∈ ℝⁿ<inf>+</inf> | <i>Ax</i> ≤ <i>b</i>}, for <i>A</i>, ∈ ℝ^{<i>m</i>×<i>n</i>}, <i>b</i> ∈ ℝ^<i>n</i>. The computational complexity of both algorithms is essentially described by <i>n</i>^<i>m</i>, which makes them especially attractive for large <i>n</i> and relatively small <i>m</i>, when the other methods with <i>O</i>(<i>m</i>^<i>n</i>) complexity fail. The methodology, which differs from previous existing methods, uses a Laplace transform technique that is well suited to the half-space representation of Ω.

Distributions for negative-feedback-regulated stochastic gene expression: dimension reduction and numerical solution of the chemical master equation.

In this work we introduce a novel approach to study biochemical noise. It comprises a simplification of the master equation of complex reaction schemes (via an adiabatic approximation) and the numerical solution of the reduced master equation. The accuracy of this procedure is tested by comparing its results with analytic solutions (when available) and with Gillespie stochastic simulations. We further employ our approach to study the stochastic expression of a simple gene network, which is subject to negative feedback regulation at the transcriptional level. Special attention is paid to the influence of negative feedback on the amplitude of intrinsic noise, as well as on the relaxation rate of the system probability distribution function to the steady solution. Our results suggest the existence of an optimal feedback strength that maximizes this relaxation rate.

On Counting Integral Points in a Convex Rational Polytope

Given a convex rational polytope ?( b) := { x ? R n+ |Ax = b}, we consider the functionb ?f( b), which counts the nonnegative integral points of ?( b). A closed form expression of its Z-transformz?F( z) is easily obtained so thatf( b) can be computed as the inverse Z-transform of F. We then provide two variants of an inversion algorithm. As a by-product, one of the algorithms provides the Ehrhart polynomial of a convex integer polytope ?. We also provide an alternative that avoids the complex integration of F( z) and whose main computational effort is to solve a linear system. This latter approach is particularly attractive for relatively small values ofm, wherem is the number of nontrivial constraints (or rows ofA).

Cycling expression and cooperative operator interaction in the trp operon of Escherichia coli

Oscillatory behaviour in the tryptophan operon of an Escherichia coli mutant strain lacking the enzyme-inhibition regulatory mechanism has been observed by Bliss et al. but not confirmed by others. This behaviour could be important from the standpoint of synthetic biology, whose goals include the engineering of intracellular genetic oscillators. This work is devoted to investigating, from a mathematical modelling point of view, the possibility that the trp operon of the E. coli inhibition-free strain expresses cyclically. For that we extend a previously introduced model for the regulatory pathway of the tryptophan operon in Escherichia coli to account for the observed multiplicity and cooperativity of repressor binding sites. Thereafter we investigate the model dynamics using deterministic numeric solutions, stochastic simulations, and analytic studies. Our results suggest that a quasi-periodic behaviour could be observed in the trp operon expression level of single bacteria.

The utility of simple mathematical models in understanding gene regulatory dynamics

Abstract In this review, we survey work that has been carried out in the attempts of biomathematicians to understand the dynamic behaviour of simple bacterial operons starting with the initial work of the 1960’s. We concentrate on the simplest of situations, discussing both repressible and inducible systems and then turning to concrete examples related to the biology of the lactose and tryptophan operons. We conclude with a brief discussion of the role of both extrinsic noise and so-called intrinsic noise in the form of translational and/or transcriptional bursting.

Simple Explicit Formula for Counting Lattice Points of Polyhedra

Given zi¾? i¾?nand Ai¾? i¾?m×n, we provide an explicit expression and an algorithm for evaluating the counting function h(y;z): = i¾? { zx| xi¾? i¾?n;Ax=y,xi¾? 0}. The algorithm only involves simple (but possibly numerous) calculations. In addition, we exhibit finitely manyfixed convex cones of i¾?nexplicitly and exclusively defined by A, such that for anyyi¾? i¾?m, h(y;z) is obtained by a simple formula that evaluates i¾? zxover the integral points of those cones only. At last, we also provide an alternative (and different) formula from a decomposition of the generating function into simpler rational fractions, easy to invert.

The Tryptophan Operon

Tryptophan is one of the 20 amino acids out of which all proteins are made. Arguably, tryptophan is the most expensive amino acid to synthesize, biochemically speaking. Perhaps, for this reason, humans and many other mammals do not have the enzymes necessary to catalyze tryptophan synthesis and instead they find this amino acid in their diet.

The Lysis-Lysogeny Switch

To talk about the lysis-lysogeny switch, we need to talk about bacteriophage (or simply phage) λ. And to talk about phage λ, we need to talk about bacteriophages (or simply phages) in general. So, let us begin with a brief historical review of the research involving these fascinating creatures.

Generic Deterministic Models of Prokaryotic Gene Regulation

The central tenet of molecular biology was put forward some half century ago, and though modified in detail still stands in its basic form. Transcription of DNA produces messenger RNA (mRNA, denoted M here). Then through the process of translation of mRNA, intermediate protein (I) is produced which is capable of controlling metabolite (E) levels that in turn can feedback and affect transcription and/or translation. A typical example would be in the lactose operon of Chap. 5 where the intermediate is β-galactosidase and the metabolite is allolactose. These metabolites are often referred to as effectors, and their effects can, in the simplest case, be either stimulatory (so called inducible) or inhibitory (or repressible) to the entire process. This scheme is often called the ‘operon concept’.

General Dynamic Considerations

The Goodwin model for operon dynamics (Goodwin, Adv Enzyme Regul 3:425–438, 1965) considers a large population of cells, each of which contains one copy of a particular operon, and we use that as a basis for discussion. We let (M, I, E) respectively denote the mRNA, intermediate protein, and effector concentrations. For a generic operon with a maximal level of transcription \(\bar{b}_{d}\) (in concentration units), the dynamics are given by Goodwin (Adv Enzyme Regul 3:425–438, 1965), Griffith (J Theor Biol 20:202–208, 1968a), Griffith (J Theor Biol 20:209–216, 1968b) Othmer (J Math Biol 3:53–78, 1976), Selgrade (SIAM J Appl Math 36:219–229, 1979) \(\begin{array}{rcl} \dfrac{dM} {dt} & =&\bar{b}_{d}\bar{\varphi }_{m}f(E) -\gamma _{M}M, \\ \dfrac{dI} {dt} & =&\beta _{I}M -\gamma _{I}I, \\ \dfrac{dE} {dt} & =&\beta _{E}I -\gamma _{E}E. \end{array}\) It is assumed here that the rate of mRNA production is proportional to the fraction of time the operator region is active, and that the rates of protein and metabolite production are proportional to the amount of mRNA and intermediate protein respectively. All three of the components (M, I, E) are subject to degradation, and the function f is as determined in Chap. 1 above.