On the Mathematical Modelling of Microbial Growth: Some Computational Aspects

We propose a new approach to the mathematical modelling of microbial growth. Our approach differs from familiar Monod type models by considering two phases in the physiological states of the microorganisms and makes use of basic relations from enzyme kinetics. Such an approach may be useful in the modelling and control of biotechnological processes, where microorganisms are used for various biodegradation purposes and are often put under extreme inhibitory conditions. Some computational experiments are performed in support of our modelling approach.* The author was partially supported by the Bulgarian NSF Project DO 02-359/2008

On the Arithmetic of Errors

An approximate number is an ordered pair consisting of a (real) number and an error bound, briefly error, which is a (real) non-negative number. To compute with approximate numbers the arithmetic operations on errors should be well-known. To model computations with errors one should suitably define and study arithmetic operations and order relations over the set of non-negative numbers. In this work we discuss the algebraic properties of non-negative numbers starting from familiar properties of real numbers. We focus on certain operations of errors which seem not to have been sufficiently studied algebraically. In this work we restrict ourselves to arithmetic operations for errors related to addition and multiplication by scalars. We pay special attention to subtractability-like properties of errors and the induced “distance-like” operation. This operation is implicitly used under different names in several contemporary fields of applied mathematics (inner subtraction and inner addition in interval analysis, generalized Hukuhara difference in fuzzy set theory, etc.) Here we present some new results related to algebraic properties of this operation.* The first author was partially supported by the Bulgarian NSF Project DO 02-359/2008 and NATO project ICS.EAP.CLG 983334

Towards an Axiomatization of Interval Arithmetic

In this paper intervals are viewed as approximate real numbers. A revised formula for interval multiplication of generalized intervals is given. This formula will be useful for further axiomatization of interval arithmetic and relevant implementations within computer algebra systems. Relations between multiplication of numbers and multiplication of errors are discussed

Analysis of Biochemical Mechanisms using Mathematica with Applications

Biochemical mechanisms with mass action kinetics are usually modeled as systems of ordinary differential equations (ODE) or bipartite graphs. We present a software module for the numerical analysis of ODE models of biochemical mechanisms of chemical species and elementary reactions (BMCSER) within the programming environment of CAS Mathematica. The module BMCSER also visualizes the bipartite graph of biochemical mechanisms. Numerical examples, including a double phosphorylation model, are presented demonstrating the scientific applications and the visualization properties of the module. ACM Computing Classification System (1998): G.4

From the Guest-Editor

The BIOMATH 2012 International Conference on Mathematical Methods and Models in Biosciences was held at the Bulgarian Academy of Sciences in Sofia, in June 17-22, 2012, http://www.biomath.bg/2012/.  We were happy to meet more than 70 participants  from twenty different  countries.   More than 40 contributions were submitted for publication in the present BIOMATH proceedings.  All submitted papers have been peer-reviewed by at least two independent anonymous reviewers. Twelve selected papers are published in the first issue of this journal. This second issue contains another ten selected contributions which will be published continuously in the electronic version of the journal.   . .

From the Guest Editor

The BIOMATH 2012 International Conference on Mathematical Methods and Models in Biosciences was held at the Academy of Sciences in Sofia, Bulgaria, in June 17–22, 2012, http://www.biomath.bg/2012/. We were happy to meet more than 70 participants from twenty different countries. More than 40 contributions were submitted for publication in the present BIOMATH proceedings

Cell Growth Models Using Reaction Schemes: Batch Cultivation

Simple structured mathematical models of bacterial cell growth are proposed. The models  involve fractions of bacterial cells related  to  their physiological phases. Reaction schemes involving the biomass of the  cell fractions, the substrate and the product are proposed in analogy to reaction schemes in enzyme kinetics. Applying the mass action law these reaction schemes  lead to  dynamical models represented by  systems of ODE's. All parameters of the models  are rate constants with clear biological or biochemical meaning.  The proposed models generalize  classical bacterial growth models and offer more flexible tools for modelling and control of biotechnological processes. In this paper the study is focused on batch cultivation models

Reaction networks reveal new links between Gompertz and Verhulst growth functions

New reaction network realizations of the Gompertz and logistic growth models are proposed. The proposed reaction networks involve an additional species interpreted as environmental resource. Some natural generalizations and modifications of the Gompertz and the logistic models, induced by the proposed networks, are formulated and discussed. In particular, it is shown that the induced dynamical systems can be reduced to one dimensional differential equations for the growth (resp. decay) species. The reaction network formulation of the proposed models suggest hints for the intrinsic mechanism of the modeled growth process and can be used for analyzing evolutionary measured data when testing various appropriate models, especially when studying growth processes in life sciences