Some multi-dimensional branching processes as motivated by a class of problems in mathematical genetics II

This paper is a continuation of a paper, “Some Multi-Dimensional Branching Processes as Motivated by a Class of Problems in Mathematical Genetics I,” by C. J. Mode, which appeared in a previous issue of theBulletin. Its purpose is two-fold; namely to discuss the mathematical existence of the model and to supply the mathematical proofs of some theorems in section two of the paper mentioned above. This paper should be read in conjunction with the previous paper. The research reported in this paper was supported in part by the United States Atomic Energy Commission, Division of Biology and Medicine Project AT(45-1)-1729.     

A theoretical approach to G-protein modulation of cellular responsiveness

 Structure and function of cells often depend critically on molecular signals arriving at their surface. There are universal mechanisms of signal transduction and signal processing across cell membranes. In this paper the mechanisms involving guanine-nucleotide regulatory proteins (“G-proteins”) and certain receptor-kinases are considered. On the basis of recent findings in molecular biology a mathematical model is developed taking into account all essential components in the biochemical network between first and second messenger. There are two coupled feedback loops inherent in this process. The model finally consists of three nonlinear equations, which are obtained from a system of originally ten equations by using conservation laws and quasi-steady state conditions. The second part of the paper contains a mathematical analysis of the model. Solutions describing the temporal development of the involved biochemical species are shown to be bounded, more specifically to remain, independent of the size of the input signal, in a bounded domain of the state space. For the situation of stationary input signals existence, uniqueness and asymptotic stability of steady states are derived. We also demonstrate biologically relevant stimulus-response properties like monotonicity and saturation effects. For temporally non-constant input signals we show numerically that the model is able to produce phenomena of hypersensitivity and desensitization which are important characteristics of cellular responsiveness. Received 18 March 1996; received in revised form 15 April 1996     

The Africanized honey bee dispersal: A mathematical zoom

A general mathematical model for population dispersal featuring long range taxis is presented and exemplified by the dispersal episode of the Africanized honey bees (Apis mellifera adansonii) throughout the American Continent. The mathematical model is a discrete-time and nonlocal model represented by an integrodifference recursion. A newtaxis concept is defined and introduced into the mathematical model by an appropriate modification of the redistribution kernel. The model is capable of predicting the natural barrier for the expansion of the Africanized honey bees in the southern part of the Continent due to low winter temperatures. It also describes a sensitive expansion velocity with respect to the quality of resources, which can explain the AHB’s astounding spread rate, by using two different kinds of population dynamics strategies, one for a resourceful environment and the other for poor regions. An erratum to this article is available at .     

Mathematical Modelling of the <Emphasis Type="Italic">Aux/IAA</Emphasis> Negative Feedback Loop

The hormone auxin is implicated in regulating a diverse range of developmental processes in plants. Auxin acts in part by inducing the Aux/IAA genes. The associated pathway comprises multiple negative feedback loops (whereby Aux/IAA proteins can repress Aux/IAA genes) that are disrupted by auxin mediating the turnover of Aux/IAA protein. In this paper, we develop a mathematical model of a single Aux/IAA negative feedback loop in a population of identical cells. The model has a single steady-state. We explore parameter space to uncover a number of dynamical regimes. In particular, we identify the ratio between the Aux/IAA protein and mRNA turnover rates as a key parameter in the model. When this ratio is sufficiently small, the system can evolve to a stable limit cycle, corresponding to an oscillation in Aux/IAA expression levels. Otherwise, the steady-state is either a stable-node or a stable-spiral. These observations may shed light on recent experimental results.     

Modelling of the regulation of the hilA promoter of type three secretion system of Salmonella enterica serovar Typhimurium

One of the most common modes of secretion of toxins in gram-negative bacteria is via the type three secretion system (TTSS), which enables the toxins to be specifically exported into the host cell. The hilA gene product is a key regulator of the expression of the TTSS located on the pathogenicity island (SPI-1) of Salmonella enterica serovar Typhimurium. It has been proposed earlier that the regulation of HilA expression is via a complex feedforward loop involving the transactivators HilD, HilC and RtsA. In this paper, we have constructed a mathematical model of regulation of hilA-promoter by all the three activators using two feedforward loops. We have modified the model to include additional complexities in regulation such as the proposed positive feedback and cross regulations of the three transactivators. Results of the various models indicate that the basic model involving two Type I coherent feedforward loops with an OR gate is sufficient to explain the published experimental observations. We also discuss two scenarios where the regulation can occur via monomers or heterodimers of the transactivators and propose experiments that can be performed to distinguish the two modes of regulator function. Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Brain and physics of many-body problems

Summary On the basis of a recent physical theory of many-body problems developped in our Institute, a model of the brain is formulated, and it is shown how some of its typical features, such as learning and memory processes, find therein a natural and simple explanation. In the Appendix a short surview of the necessary mathematical formalism is finally given.A brief outline of the model presented in this paper was unformally given by one of the authors (L.M.R.) at the I.S.P. Neurosciences Research Program; Boulder Colorado, in July 1966.This work has been sponsored in part by: Air Force Avionics Lab., Research and Technology Div., Wright Patterson Air Force, Air Force System Commend-U.S.A.F.-Government of United States of America. Contract No. AF 33(615)-2786.     

A model of the role of natural killer cells in immune surveillance — I

Summary The theory of immune surveillance of Thomas and Burnet stated in part that antigenic differences between neoplastic and normal cells provide the stimulus for their destruction by cells of the immune system. Burnet pointed to the T lymphocyte as the cell which mediated this surveillance. The existence of some form of surveillance in cases of no T lymphocyte functioning presents the possibility that surveillance, if present at all, is mediated by non T cells.Cells identified as naturally cytotoxic killer (NK) cells appear to have properties required of a surveillance effector population. This paper utilizes properties of NK cells and the effects of interferon on this population to construct a mathematical model of the characteristics that an NK cell surveillance would have. A two level theory of immune surveillance is proposed.This research has been supported in part by the National Science Foundation under grant #NSF-Eng. 7904852     

A mathematical analysis of small mammal populations

Populations of Microtus montanus, the montane vole, have been extensively studied. It is known that their reproductive activity is closely linked to the availability of the chemicals in growing plants. We use a mathematical model here to study how the length of the vegetative season and the natural reproduction rhythm of voles are involved in the long term dynamics of the population numbers. In particular, we use data obtained from Timpie Springs, Utah, and from Jackson Hole, Wyoming, to formulate a model. The novelty of this model is its use of littering curves that highlight the temporally discrete nature of vole reproduction. The model shows how the timing of the vegetative season can influence vole population sizes.This work was supported in part by NSF Grant No. MCS-82-08986 and DMS85-14000 (FCH) and by the University of Utah where Dr. Murphy was a visitor     

Macrophage response to <Emphasis Type="Italic">Mycobacteriumtuberculosis</Emphasis> infection

The immune response to Mycobacterium tuberculosis (Mtb) infection is the formation of multicellular lesions, or granolomas, in the lung of the individual. However, the structure of the granulomas and the spatial distribution of the immune cells within is not well understood. In this paper we develop a mathematical model investigating the early and initial immune response to Mtb. The model consists of coupled reaction-diffusion-advection partial differential equations governing the dynamics of the relevant macrophage and bacteria populations and a bacteria-produced chemokine. Our novel application of mathematical concepts of internal states and internal velocity allows us to begin to study this unique immunological structure. Volume changes resulting from proliferation and death terms generate a velocity field by which all cells are transported within the forming granuloma. We present numerical results for two distinct infection outcomes: controlled and uncontrolled granuloma growth. Using a simplified model we are able to analytically determine conditions under which the bacteria population decreases, representing early clearance of infection, or grows, representing the initial stages of granuloma formation.     

Interaction of Tumor with Its Micro-environment: A Mathematical Model

This paper is concerned with early development of transformed epithelial cells (TECs) in the presence of fibroblasts in the tumor micro-environment. These two types of cells interact by means of cytokines such as transforming growth factor (TGF-β) and epidermal growth factor (EGF) secreted, respectively, by the TECs and the fibroblasts. As this interaction proceeds, TGF-β induces fibroblasts to differentiate into myofibroblasts which secrete EGF at a larger rate than fibroblasts. We monitor the entire process in silico, in a setup which mimics experiments in a Tumor Chamber Invasion Assay, where a semi-permeable membrane coated by extracellular matrix (ECM) is placed between two chambers, one containing TECs and another containing fibroblasts. We develop a mathematical model, based on a system of PDEs, that includes the interaction between TECs, fibroblasts, myofibroblasts, TGF-β, and EGF, and we show how model parameters affect tumor progression. The model is used to generate several hypotheses on how to slow tumor growth and invasion. In an Appendix, it is proved that the mathematical model has a unique global in-time solution.     

A mathematical model for quorum sensing in Pseudomonas aeruginosa

The bacteria Pseudomonas aeruginosa use the size and density of their colonies to regulate the production of a large variety of substances, including toxins. This phenomenon, called quorum sensing, apparently enables colonies to grow to sufficient size undetected by the immune system of the host organism. In this paper, we present a mathematical model of quorum sensing in P. aeruginosa that is based on the known biochemistry of regulation of the autoinducer that is crucial to this signalling mechanism. Using this model we show that quorum sensing works because of a biochemical switch between two stable steady solutions, one with low levels of autoinducer and one with high levels of autoinducer.     

Diffusion regulated growth characteristics of a spherical prevascular carcinoma

Recently a mathematical model of the prevascular phases of tumor growth by diffusion has been investigated (S. A. Maggelakis and J. A. Adam,Math. Comput. Modeling, in press). In this paper we examine in detail the results and implications of that mathematical model, particularly in the light of recent experimental work carried out on multicellular spheroids. The overall growth characteristics are determined in the present model by four parameters:Q, γ, b, andδ, which depend on information about inhibitor production rates, oxygen consumption rates, volume loss and cell proliferation rates, and measures of the degree of non-uniformity of the various diffusion processes that take place. The integro-differential growth equation is solved for the outer spheroid radiusR ₀(t) and three related inner radii subject to the solution of the governing time-independent diffusion equations (under conditions of diffusive equilibrium) and the appropriate boundary conditions. Hopefully, future experimental work will enable reasonable bounds to be placed on parameter values referred to in this model: meanwhile, specific experimentally-provided initial data can be used to predict subsequent growth characteristics ofin vitro multicellular spheroids. This will be one objective of future studies.     

Modeling retrovirus production for gene therapy. 1. Determination Of optimal bioreaction mode and harvest strategy

Although retroviruses are a promising tool for gene therapy, there are two major problems limiting the establishment of viable industrial processes: retrovirus stability and low final yield in the supernatant. This fact emphasizes the need for an effective process optimization, not only at a genetic level but also at a bioprocess engineering level. In part 1 of this paper a mathematical model was developed to optimize the bioreaction yield by determining the best retrovirus harvest strategy in perfusion cultures. PA317 cells producing recombinant retroviruses were used to develop and test this model. Cell culture was performed in stirred tanks using porous supports. The parameters of the proposed model were experimentally determined for batch and perfusion cultures at 32 and 37 degrees C both with and without additives to enhance production; the model was then validated. This model allowed the determination of the optimal values of all operational variables included: batch and perfusion duration and perfusion rate. The highest productivity (2682 virus cm(-)(3) h(-)(1)) was obtained under the following conditions: batch at 37 degrees C for 53 h followed by perfusion at 32 degrees C for 23 h with a perfusion rate of 0.107 h(-)(1). This value was 3.5-fold higher than the best result obtained in batch cultures for the same conditions of titer and quality. A sensitivity analysis of the parameters showed that the parameters that affect most the final productivity depend on the bioreaction phase: cell growth in batch culture and production and product degradation in perfusion culture. In part 2 of this paper, this model is extended to the first step of downstream processing, and the addition of further steps to the process is discussed in order to achieve global process optimization.     

Multiregional Periodic Matrix for Modeling the Population Dynamics of Sardine (<Emphasis Type="Italic">Sardina pilchardus</Emphasis>) Along the Moroccan Atlantic Coast: Management Elements for Fisheries

In this paper, we present a deterministic time discrete mathematical model based on multiregional periodic matrices to describe the dynamics of Sardina pilchardus in the Central Atlantic area of the Moroccan coast. This model deals with two stages (immature and mature) and three spatial zones where sardines are supposed to migrate from one zone to another. The population dynamics is described by an autonomous recurrence equation N(t + 1) = A.N(t), where A is a positive matrix whose entries are estimated using data collected during biannual acoustic surveys carried out from 2001 to 2003 onboard the Norwegian research vessel “Dr Fridtjof Nansen”. The dominant eigenvalue λ of A that gives the long-term growth rate of fish population is smaller than one. This agrees with the stock decrease observed in the data collected. We show that λ is highly sensitive to the recruitment rate and much less sensitive to the reproduction rate. These results can clearly be used to define an efficient scenario in order to fight for instance against a stock decrease.     

The Effect of Chemical Information on the Spatial Distribution of Fruit Flies: II Parameterization,Calibration, and Sensitivity

In a companion paper (Lof et al., in Bull. Math. Biol., 2008), we describe a spatio-temporal model for insect behavior. This model includes chemical information for finding resources and conspecifics. As a model species, we used Drosophila melanogaster, because its behavior is documented comparatively well. We divide a population of Drosophila into three states: moving, searching, and settled. Our model describes the number of flies in each state, together with the concentrations of food odor and aggregation pheromone, in time and in two spatial dimensions. Thus, the model consists of 5 spatio-temporal dependent variables, together with their constituting relations. Although we tried to use the simplest submodels for the separate variables, the parameterization of the spatial model turned out to be quite difficult, even for this well-studied species. In the first part of this paper, we discuss the relevant results from the literature, and their possible implications for the parameterization of our model. Here, we focus on three essential aspects of modeling insect behavior. First, there is the fundamental discrepancy between the (lumped) measured behavioral properties (i.e., fruit fly displacements) and the (detailed) properties of the underlying mechanisms (i.e., dispersivity, sensory perception, and state transition) that are adopted as explanation. Detailed quantitative studies on insect behavior when reacting to infochemicals are scarce. Some information on dispersal can be used, but quantitative data on the transition between the three states could not be found. Second, a dose-response relation as used in human perception research is not available for the response of the insects to infochemicals; the behavioral response relations are known mostly in a qualitative manner, and the quantitative information that is available does not depend on infochemical concentration. We show how a commonly used Michaelis–Menten type dose-response relation (incorporating a saturation effect) can be adapted to the use of two different but interrelated stimuli (food odors and aggregation pheromone). Although we use all available information for its parameterization, this model is still overparameterized. Third, the spatio-temporal dispersion of infochemicals is hard to model: Modeling turbulent dispersal on a length scale of 10 m is notoriously difficult. Moreover, we have to reduce this inherently three-dimensional physical process to two dimensions in order to fit in the two-dimensional model for the insects. We investigate the consequences of this dimension reduction, and we demonstrate that it seriously affects the parameterization of the model for the infochemicals. In the second part of this paper, we present the results of a sensitivity analysis. This sensitivity analysis can be used in two manners: firstly, it tells us how general the simulation results are if variations in the parameters are allowed, and secondly, we can use it to infer which parameters need more precise quantification than is available now. It turns out that the short term outcome of our model is most sensitive to the food odor production rate and the fruit fly dispersivity. For the other parameters, the model is quite robust. The dependence of the model outcome with respect to the qualitative model choices cannot be investigated with a parameter sensitivity analysis. We conclude by suggesting some experimental setups that may contribute to answering this question.     

Modelling the biological invasion of Carcinus maenas (the European green crab)

This paper proposes a system of integro-difference equations to model the spread of Carcinus maenas, commonly called the European green crab, that causes severe damage to coastal ecosystems. A model with juvenile and adult classes is first studied. Here, standard theory of monotone operators for integro-difference equations can be applied and yields explicit formulas for the asymptotic spreading speeds of the juvenile and adult crabs. A second model including an infected class is considered by introducing a castrating parasite Sacculina carcini as a biological control agent. The dynamics are complicated and simulations reveal the occurrence of periodic solutions and stacked fronts. In this case, only conjectures can be made for the asymptotic spreading speeds because of the lack of mathematical theory for non-monotone operators. This paper also emphasizes the need for mathematical studies of non-monotone operators in heterogeneous environments and the existence of stacked front solutions in biological invasion models.     

Role of infection on the stability of a predator-prey system with several response functions--a comparative study

In this paper, we have proposed and analyzed a mathematical model of an infected predator-prey system with different predators' functional response. The existence and uniqueness of solutions are established and solutions are shown to be uniformly bounded for all nonnegative initial values. Our overall mathematical and biological studies reveal that if the prey population is infected by a lethal disease, coexistence of all three species (i.e. host, parasite and predator) for any of three functional responses is never possible but different interesting dynamical behaviors are possible by varying two key parameters viz. the rate of infection and the attack rate on susceptible prey. Interplay between these two factors yields a diverse array of biologically relevant behavior, including switching of stability, extinction and oscillations.     

Recombination between heterologous linear and circular mitochondrial plasmids in the fungus Neurospora

A strain of Neurospora intermedia from China contains five prominent extragenomic mitochondrial plasmids: three linear elements called zhisi plasmids, and two circular plasmids, Harbin-1 and -2. In one subculture, levels of four plasmids (all three zhisis and Harbin-1) fell to undetectable values and two novel linear plasmids appeared, Harbin-L and -L2, as well as a new small circular plasmid, Harbin-0.9. Cross-hybridization of restriction fragments and DNA sequencing showed that the Harbin-L plasmid was composed of parts of the circular Harbin-1 plasmid and of one of the linear zhisi plasmids. A model is presented in which the Harbin-1 and zhisi plasmids are present within the same mitochondrion, and crossovers at two separate 7 by sites of sequence identity effectively insert part of the circular Harbin-1 DNA into a zhisi linear plasmid, simultaneously deleting part of the zhisi element. The small plasmid Harbin-0.9 is a fragment of the Har-1 plasmid, and seems to be another product of the recombination process that created Har-L. Recombination of this type could have contributed to the wide array of mitochondrial plasmids found in natural populations of Neurospora.     

Maintenance of heterocyst patterning in a filamentous cyanobacterium

In the absence of sufficient combined nitrogen, some filamentous cyanobacteria differentiate nitrogen-fixing heterocysts at approximately every 10th cell position. As cells between heterocysts grow and divide, this initial pattern is maintained by the differentiation of a single cell approximately midway between existing heterocysts. This paper introduces a mathematical model for the maintenance of the periodic pattern of heterocysts differentiated by Anabaena sp. strain PCC 7120 based on the current experimental knowledge of the system. The model equations describe a non-diffusing activator (HetR) and two inhibitors (PatS and HetN) that undergo diffusion in a growing one-dimensional domain. The inhibitors in this model have distinct diffusion rates and temporal expression patterns. These unique aspects of the model reflect recent experimental findings regarding the molecular interactions that regulate patterning in Anabaena. Output from the model is in good agreement with both the temporal and spatial characteristics of the pattern maintenance process observed experimentally.