Current Developments in Mathematical Biology

INTRODUCTION.

CONTENTS.

Previously we have explained the abstract concept we call 'aqueous computing' and illustrated it with concrete wet lab results. Here, we explore the use of methylase enzymes to 'write' on double-stranded DNA molecules at sites where restriction enzymes will cut if, and only if, the sites have not previously been methylated. A site represents the bit zero (False, F) if the site has been methylated and the bit one (True, T) if it has not been methylated. 'Reading' is done by attempting a cut at each of the sites. We found 8 commercially available methylases and 8 corresponding restriction enzymes that would not cut after the action of one of the methylases. We were able to confirm that methylation by each of these 8 enzymes individually blocked cleavage only by the restriction enzyme associated with that site and not any other enzyme. We then used these enzymes to approach a 3-variable, 4-clause satisfiability (SAT) problem using either plasmid DNA (pBluescript) or PCR product made from the region containing the restriction enzyme sites on the plasmid. Pairs of methylases were defined to represent each of the states of the operators p, q and r, one methylase for p and another for p', etc. We methylated the DNA in parallel at the two sites so either the p site was methylated (making p false) or the p' site was methylated (making p' false). We did that for the other two variables as well to create a set of logically consistent DNA fragments. Then we applied the 4 clauses using restriction enzymes to cut DNA fragments that did not satisfy them. At the end, we found evidence for intact DNA indicating an answer satisfying all of the clauses. To confirm the state of each of the Boolean operators, we used cleavage by the appropriate restriction enzyme. We found in the computation with both the plasmid and the PCR product, one site pair to show false in both sites; q and q', for instance. This should not be possible. We suspected incomplete cutting during the clauses by one of these restriction enzymes, specifically BssHII. In summary, we did successfully show the usefulness of DNA methylation in a scheme to do a mathematical computation. Thus, we have added to our arsenal of potential methods of performing DNA computing in the aqueous style.

Boolean networks are models of genetic regulatory networks. S. Kauffman based many of his claims about spontaneous self-organization in complex systems on simulations of randomly constructed Boolean networks. Some of these claims are precise mathematical statements. We analyze these statements using combinatorial methods and show that there is partial agreement with some of Kauffman's conclusions, but in other cases there is disagreement. Our key finding is an algebraic parameter that determines the likelihood of order behaviour in a random Boolean network. There is a threshold such that when the parameter is less than the threshold, ordered behavior is prevalent, and when it is greater than the threshold, chaotic behavior is highly likely. When the parameter equals the threshold, some forms of ordered behavior persist, but others do not.

No abstract received.

We describe a basic mathematical model of phagocytosis. We outline the ideas involved in building the model as well as possible directions for further developments and refinements of the current model.

Although it is well established that T cell populations dramatically fluctuate when the immune system responds to many diseases, many specifics of how T cell activation occurs are not understood. We develop a model that tracks T cell interactions with dendritic cells, which is a crucial process for activating adaptive immune responses. We assume that the duration of stimulation by dendritic cells in the lymph node determines the extent of T cell differentiation. In this paper, we first describe the general partial differential equation model with justifications for the modeling choices. Second, we present a few highlights of the mathematical analysis of the model with certain simplifying assumptions. Finally, we discuss some simulations of the full model and interpret the biological predictions the model suggests.

This expository paper provides an introduction to computational methods for solving continuous models of certain biological systems. These systems have important behavior in space and within the age distribution of individuals. Very often, the spatial behavior depends critically on the age structure of the population. The different spatial scales induce different time scales in the problem, whereas age and time advance together.

Age- and space-structured multiscale systems arise in a wide variety of biological problems ranging from multicellular and tissue-level phenomena to problems in ecology and evolutionary biology. We describe a general modeling framework used to represent such biological systems. We then discuss computational methods used to solve the model equations, beginning with a treatment of the moving-grid Galerkin method used to decouple age and time while they advance together, introduce how this discretization in age works with discretizations in time and space, and then review how more complicated nonlinear problems would be treated. We close by presenting two example systems, swarm-colony development of the bacteria Proteus mirabilis and tumor invasion, which differ in some important respects from the general model, but which are effectively treated by the computational methods presented in this paper.

A simple model of phytoplankton-zooplankton interaction with a periodic input nutrient is presented. The model is then used to study a nutrient-plankton interaction with a toxic substance that inhibits the growth rate of phytoplankton. The effects of the toxin upon the existence, magnitude, and stability of the periodic solutions are discussed.

Strain replacement is the effect of substitution of a strain of higher prevalence in the population with another. Differential effectiveness of the vaccines is thought to be the mechanism responsible for the replacement effect. Recent theoretical study shows that differential effectiveness of the vaccine may not be necessary and other trade-off mechanisms can lead to it even when the vaccine is "perfect". We suggest that the mechanism of strain replacement is the reciprocal effect of vaccination on the fitness of the strains as measured by their invasion reproduction numbers. This mechanism is responsible for the substitution of one strain with another to occur both when the vaccine is perfect and when it is imperfect. We review various well-known trade-off mechanisms and investigate whether they lead to replacement effect in conjunction with "perfect" vaccination. We find that in contrast to imperfect vaccination which leads to replacement of a strain with larger intrinsic reproduction number with a strain with a lower intrinsic reproduction number, "perfect" vaccination seems to have opposite effect on the intrinsic reproduction numbers.