Optimal strategies in immunology III. The IgM-IgG switch

Kinetics of Hepatitis E Virus Infections in Asymptomatic Persons

To determine the kinetics of hepatitis E virus (HEV) in asymptomatic persons and to evaluate viral load doubling time and half-life, we retrospectively tested samples retained from 32 HEV RNA-positive asymptomatic blood donors in Germany. Close-meshed monitoring of viral load and seroconversion in intervals of ≈4 days provided more information about the kinetics of asymptomatic HEV infections. We determined that a typical median infection began with PCR-detectable viremia at 36 days and a maximum viral load of 2.0 × 104 IU/mL. Viremia doubled in 2.4 days and had a half-life of 1.6 days. HEV IgM started to rise on about day 33 and peaked on day 36; IgG started to rise on about day 32 and peaked on day 53. Although HEV IgG titers remained stable, IgM titers became undetectable in 40% of donors. Knowledge of the dynamics of HEV viremia is useful for assessing the risk for transfusion-transmitted hepatitis E.

Trends in mathematical modeling of host-pathogen interactions

Pathogenic microorganisms entail enormous problems for humans, livestock, and crop plants. A better understanding of the different infection strategies of the pathogens enables us to derive optimal treatments to mitigate infectious diseases or develop vaccinations preventing the occurrence of infections altogether. In this review, we highlight the current trends in mathematical modeling approaches and related methods used for understanding host-pathogen interactions. Since these interactions can be described on vastly different temporal and spatial scales as well as abstraction levels, a variety of computational and mathematical approaches are presented. Particular emphasis is placed on dynamic optimization, game theory, and spatial modeling, as they are attracting more and more interest in systems biology. Furthermore, these approaches are often combined to illuminate the complexities of the interactions between pathogens and their host. We also discuss the phenomena of molecular mimicry and crypsis as well as the interplay between defense and counter defense. As a conclusion, we provide an overview of method characteristics to assist non-experts in their decision for modeling approaches and interdisciplinary understanding.

Plant protein diet suppressed immune function by inhibiting spiral valve intestinal mucosal barrier integrity, anti-oxidation, apoptosis, autophagy and proliferation responses in amur sturgeon (Acipenser schrenckii)

An 8-week growth trial was conducted to investigate the effects of replacing dietary fishmeal with a plant protein blend on the growth performance, mucosal barrier integrity and the related regulation mechanism in Amur Sturgeon (Acipenser schrenckii) with initial weight of 87.48 g. Three isonitrogenous and isoenergetic diets were prepared. A basal diet containing 540 g/kg fishmeal (P0), whereas the other two diets were formulated by replacing 50% and 100% of FM with plant protein blend (soybean protein concentrate and cottonseed protein concentrate), and named as P50 and P100, respectively. Although essential amino acids, fatty acids, and available phosphorus had been balanced according to the nutrient requirement of sturgeon, compared with the fish of P0 and P50, the full plant protein diet (P100) significantly reduced growth performance and survival, and accompanied with serious spiral valve intestinal (SVI) damage. The increased tissue necrosis and failed responses in anti-oxidation, programming apoptosis, autophagy and cell proliferation system were regulated by inhibiting ERK1 phosphorylation, which indicated that SVI hypoimmunity and functional degradation were the main reasons for the high mortality and low utilization ability of plant protein in Amur sturgeon.

Mathematical modeling of within-host Zika virus dynamics

Recent Zika virus outbreaks have been associated with severe outcomes, especially during pregnancy. A great deal of effort has been put toward understanding this virus, particularly the immune mechanisms responsible for rapid viral control in the majority of infections. Identifying and understanding the key mechanisms of immune control will provide the foundation for the development of effective vaccines and antiviral therapy. Here, we outline a mathematical modeling approach for analyzing the within-host dynamics of Zika virus, and we describe how these models can be used to understand key aspects of the viral life cycle and to predict antiviral efficacy.

Fighting Cancer with Mathematics and Viruses

After decades of research, oncolytic virotherapy has recently advanced to clinical application, and currently a multitude of novel agents and combination treatments are being evaluated for cancer therapy. Oncolytic agents preferentially replicate in tumor cells, inducing tumor cell lysis and complex antitumor effects, such as innate and adaptive immune responses and the destruction of tumor vasculature. With the availability of different vector platforms and the potential of both genetic engineering and combination regimens to enhance particular aspects of safety and efficacy, the identification of optimal treatments for patient subpopulations or even individual patients becomes a top priority. Mathematical modeling can provide support in this arena by making use of experimental and clinical data to generate hypotheses about the mechanisms underlying complex biology and, ultimately, predict optimal treatment protocols. Increasingly complex models can be applied to account for therapeutically relevant parameters such as components of the immune system. In this review, we describe current developments in oncolytic virotherapy and mathematical modeling to discuss the benefit of integrating different modeling approaches into biological and clinical experimentation. Conclusively, we propose a mutual combination of these research fields to increase the value of the preclinical development and the therapeutic efficacy of the resulting treatments.

Simulating Immune Interference on the Effect of a Bivalent Glycoconjugate Vaccine against Haemophilus influenzae Serotypes "a" and "b"

Objective. We sought to evaluate the immune responses to a bivalent Haemophilus influenzae glycoconjugate vaccine against serotypes “a” (Hia) and “b” (Hib) in the presence of the preexisting immunity to Hib. Methods. We developed a stochastic simulation model of humoral immune response to investigate the antigenic challenge of a bivalent combined glycoconjugate vaccine and a bivalent unimolecular glycoconjugate vaccine. We compared simulation outcomes in the absence of any preexisting immunity with an already primed immune response having specific memory B cells and/or anti-Hib antibodies. Results. The simulation results show that the preexisting immune responses to Hib or carrier protein (CP) may significantly impede the production of anti-Hia antibodies by a unimolecular vaccine. In contrast, the production of anti-Hia antibodies using a combined vaccine is inhibited only in the presence of CP immune responses. Conclusions. Preexisting immunity to Hib and CP may play a critical role in the development of immune responses against Hia or Hib using bivalent combined and unimolecular vaccine formulations. Our results suggest that a bivalent combined glycoconjugate vaccine with a carrier protein not previously used in Hib conjugate vaccines may be an effective formulation for generating immune responses to protect against both Hib and Hia infections.

Extension and Justification of Quasi-Steady-State Approximation for Reversible Bimolecular Binding

The quasi-steady-state approximation (QSSA) is commonly applied in chemical kinetics without rigorous justification. We provide details of such a justification in the ubiquitous case of reversible two-step bimolecular binding in which molecules as an intermediate step of the reaction form a transient complex. First, we justify QSSA in the regime that agrees with the results in the literature and is characterized by max{R₀, L₀} ≪ K(m). Here, R₀ and L₀ are the initial concentrations of reacting receptor and ligand, respectively, and K(m) is the Michaelis constant. We also validate QSSA under an alternative condition that can be viewed as partially irreversible binding, and it does not require a tight bound on R₀ and L₀ but rather requires k₂ + k₋₂ ≪ k₋₁. Here, k₋₁ is the rate constant of decomposition of the transient complex to the ligand and the receptor, and k₂ and k₋₂ are the forward and the reverse rate constants of transformation of the complex to the product, respectively. Furthermore, we provide arguments that QSSA can also be accurate in a regime when max{R₀, L₀} ≈ K(m) and k₂ + k₋₂ ≈ k₋₁ if |R₀ - L₀| ≪ K(m). The derived conditions may be of practical use as they provide weaker requirements for the validity of QSSA compared to the existing results.

A shooting algorithm for complex immunodominance control problems

Although T cells are able to recognize a wide variety of target peptides, they are often strongly focused on a few of the peptides and leave the rest of them unattended. This phenomenon of strongly biased immune response is known as immunodominance. Mathematically, an immunodominance problem can be formulated using optimal control principles as a two-point boundary-value problem. The solution of this problem is challenging especially when the control variables are bounded. In this work, we develop a numerical algorithm based on the shooting technique for bounded optimal control problems. The algorithm is applied to a group of immunodominance problems. Numerical simulations reveal that the immune system selects either a broad or a specific strategy of immunodominance based on different optimization goals. The shooting algorithm can also be utilized to solve other complex optimal control problems.

Autoantibodies to folate receptor during pregnancy and neural tube defect risk

Periconceptional folic acid can reduce the occurrence of neural tube defects (NTDs) by up to 70%, and autoantibodies for folate receptors (FRs) have been observed in serum from women with a pregnancy complicated by an NTD. This population-based cohort study has examined serum from pregnant mothers for autoantibodies to FRs, antibodies to bovine folate binding protein (FBP), and inhibition of folic acid binding to FR and FBP in association with NTD risk. The mid-gestational maternal serum specimens used for this study were collected during the 15-18th week of pregnancy. Samples were obtained from the California Birth Defects Monitoring Program; 29 mothers had a pregnancy complicated by spina bifida and 76 mothers had unaffected children. The presence of IgG and IgM antibodies to human FR, bovine FBP, and inhibition of folic acid binding to FR and FBP was determined. Higher activity of IgM to FBP in cases verses controls was observed (P=0.04). Higher activity of IgM and IgG autoantibodies to FR was observed (P<0.001 and P=0.04, respectively). Risk estimates at two standard deviations above average control antibody concentrations were OR=2.07 (CI=1.02, 4.06) for anti-FBP IgM, OR=2.15 (CI=1.02, 4.69) for anti-FR IgG and OR=3.19 (CI=1.47, 6.92) for anti-FR IgM. These data support the hypothesis that high titers of antibodies and blocking of folic acid binding to FRs by maternal serum should be regarded as risk factors for NTDs.

Tentacle probes: eliminating false positives without sacrificing sensitivity

The majority of efforts to increase specificity or sensitivity in biosensors result in trade-offs with little to no gain in overall accuracy. This is because a biosensor cannot be more accurate than the affinity interaction it is based on. Accordingly, we have developed a new class of reagents based on mathematical principles of cooperativity to enhance the accuracy of the affinity interaction. Tentacle probes (TPs) have a hairpin structure similar to molecular beacons (MBs) for enhanced specificity, but are modified by the addition of a capture probe for increased kinetics and affinity. They produce kinetic rate constants up to 200-fold faster than MB with corresponding stem strengths. Concentration-independent specificity was observed with no false positives at up to 1 mM concentrations of variant analyte. In contrast, MBs were concentration dependent and experienced false positives above 3.88 μM of variant analyte. The fast kinetics of this label-free reagent may prove important for extraction efficiency, hence sensitivity and detection time, in microfluidic assays. The concentration-independent specificity of TPs may prove extremely useful in assays where starting concentrations and purities are unknown as would be the case in bioterror or clinical point of care diagnostics.

The evolution of mathematical immunology

The types of mathematical models used in immunology and their scope have changed drastically in the past 10 years. Classical models were based on ordinary differential equations (ODEs), difference equations, and cellular automata. These models focused on the 'simple' dynamics obtained between a small number of reagent types (e.g. one type of receptor and one type of antigen or two T-cell populations). With the advent of high-throughput methods, genomic data, and unlimited computing power, immunological modeling shifted toward the informatics side. Many current applications of mathematical models in immunology are now focused around the concepts of high-throughput measurements and system immunology (immunomics), as well as the bioinformatics analysis of molecular immunology. The types of models have shifted from mainly ODEs of simple systems to the extensive use of Monte Carlo simulations. The transition to a more molecular and more computer-based attitude is similar to the one occurring over all the fields of complex systems analysis. An interesting additional aspect in theoretical immunology is the transition from an extreme focus on the adaptive immune system (that was considered more interesting from a theoretical point of view) to a more balanced focus taking into account the innate immune system also. We here review the origin and evolution of mathematical modeling in immunology and the contribution of such models to many important immunological concepts.

A primer on dynamic optimization and optimal control in pharmacoeconomics

Pharmacoeconomic analyses employ a wide range of techniques and methods to help societies allocate scarce health-care resources wisely, fairly, and efficiently. Techniques such as dynamic optimization and optimal control, however, have yet to be exploited by this field. Although control theory has a long history in mathematical biology and disease management, its application to economic costs in these disciplines has not yet been explored. Pharmacoeconomics therefore may offer a particularly promising starting point because of the emphasis this field places on the economic perspective. Although challenges may exist to implementing these techniques in practice (at least in some settings), there will nevertheless be value to considering the dynamic perspective these techniques offer, which requires thinking more critically about the optimal allocation of scare health-care resources over time. Therefore, our article serves as a primer to introduce this dynamic perspective from an economic standpoint within the context of two examples of treating of hyperlipidemia.

Determination of the immunoreactivity of radiolabeled monoclonal antibodies: a theoretical analysis

Conjugates of monoclonal antibodies with radioactive nuclides are frequently used in radioimmunotherapy. Because the conjugation procedure may adversely alter the antibody, the immunoreactivity of the antibody must be determined. In this study, we expand the mathematical description of cell-binding assays for the determination of immunoreactivity to include the amount of unlabeled antibody. We describe the relevant equations and constraints and develop a mathematical description of the underlying measurements. The dependence on antibody, on antigen or cell concentration, on the dissociation constant, and on the specific activity is taken into account. The functional dependence of bound labeled antibody on the experimental parameters is given, and the consequences for parameter estimation are discussed. In order to correctly interpret the experimental data for the immunoreactivity, one requires a mathematical model without unnecessary assumptions together with an appropriate nonlinear regression fit.

On the validity of the quasi-steady state approximation of bimolecular reactions in solution

Two-step binding kinetics are extensively used to study the relative importance of diffusion in biochemical reactions. Classical analysis of this problem assumes ad hoc that the encounter complex is at quasi-steady state (QSS). Using scaling arguments we derive a criterion for the validity of this assumption in the limit of irreversible product formation. We find that the QSS approximation (QSSA) of two-step binding is only valid if the total ligand and receptor concentrations are much smaller than (k2+k-1)/k1, where k1 and k-1 are, respectively, the forward and reverse diffusion encounter rate constants and k2 is the chemical association rate constant. This criterion can be shown to imply that the average time between encounters is much longer than the half-life of the encounter complex and also guarantees that the concentration of the encounter complex is negligible compared to the reactant and product concentrations. Numerical examples of irreversible and reversible cases corroborate our analysis and illustrate that the QSS may be invalid even if k-2<<k2. Our analysis of the irreversible case is shown to carry through to the more rigorous framework of the Smoluchowski theory of diffusion-controlled reactions. This work underscores the need for exercising greater caution in invoking the QSSA.

Stochastic optimal therapy for enhanced immune response

Therapeutic enhancement of humoral immune response to microbial attack is addressed as the stochastic optimal control of a dynamic system. Without therapy, the modeled immune response depends upon the initial concentration of pathogens in a simulated attack. Immune response can be augmented by agents that kill the pathogen directly, that stimulate the production of plasma cells or antibodies, or that enhance organ health. Using a generic mathematical model of immune response to the infection (i.e., of the dynamic state of the system), previous papers demonstrated optimal (open-loop) and neighboring-optimal (closed-loop) control solutions that defeat the pathogen and preserve organ health, given initial conditions that otherwise would be lethal [Optimal Contr. Appl. Methods 23 (2002) 91, Bioinformatics 18 (2002) 1227]. Therapies based on separate and combined application of the agents were derived by minimizing a quadratic cost function that weighted both system response and drug usage, providing implicit control over harmful side effects. Here, we focus on the effects that corrupted or incomplete measurements of the dynamic state may have on neighboring-optimal feedback control. Imperfect measurements degrade the precision of feedback adjustments to therapy; however, optimal state estimation allows the feedback strategy to be implemented with incomplete measurements and minimizes the expected effects of measurement error. Complete observability of the perturbed state for this four state example is provided by measurement of four of the six possible pairs of two variables, either set of three variables, or all four variables. The inclusion of state estimation extends the applicability of optimal control theory for developing new therapeutic protocols to enhance immune response.

Dynamic optimization of host defense, immune memory, and post-infection pathogen levels in mammals

When attacked by pathogens, higher vertebrates produce specific immune cells that fight against them. We here studied the host's optimal schedule of specific immune cell production. The damage caused by the pathogen increases with the pathogen amount in the host integrated over time. On the other hand, there is also a cost incurred by the production of specific immune cells, not only in terms of the energy needed to produce and maintain the cells, but also with respect to damages sustained by the host's body as a result of immune activity. The optimal strategy of the host is the one that minimizes the total cost, defined as a weighted sum of the damage caused by pathogens and the costs caused by the specific immune cells. The problem is solved by using Pontryagin's maximum principle and dynamic programming. The optimal defense schedule is typically as follows: In the initial phase after infection, immune cells are produced at the fastest possible rate. The amount of pathogen increases temporarily but is eventually suppressed. When the pathogen amount is suppressed to a sufficiently low level, the immune cell number decreases and converges to a low steady level, which is maintained by alternately switching between fastest production and no production. We examine the effect of time delay required to have fully active immune cells by comparing cases with different number of rate limiting steps before producing immune cells. We examine the effect of the duration of time (time delay) required before full-scale production of active immune cells by comparing cases with different numbers of rate-limiting steps before immune-cell production. We also discuss the role of immune memory based on the results of the optimal immune reaction.

Cross-linking reconsidered: binding and cross-linking fields and the cellular response

We analyze a model for the reversible cross-linking of cell surface receptors by a collection of bivalent ligands with different affinities for the receptor as would be found in a polyclonal anti-receptor serum. We assume that the amount of cross-linking determines, via a monotonic function, the rate at which cells become activated and divide. In addition to the density of receptors on the cell surface, two quantities, the binding field and the cross-linking field, are needed to characterize the cross-linking curve, i.e., the equilibrium concentration of cross-linked receptors plotted as a function of the total ligand site concentration. The binding field is the sum of all ligand site concentrations weighted by their respective binding affinities, and the cross-linking field is the sum of all ligand site concentrations weighted by the product of their respective binding and cross-linking affinity and the total receptor density. Assuming that the cross-linking affinity decreases if the binding affinity decreases, we find that the height of the cross-linking curve decreases, its width narrows, and its center shifts to higher ligand site concentrations as the affinities decrease. Moreover, when we consider cross-linking-induced proliferation, we find that there is a minimum cross-linking affinity that must be surpassed before a clone can expand. We also show that under many circumstances a polyclonal antiserum would be more likely than a monoclonal antibody to lead to cross-linking-induced proliferation.

Heterogeneity and probabilistic binding contributions to receptor-mediated cell detachment kinetics

Biospecific cell adhesion is mediated by receptor-ligand bonds. Early theoretical work presented a deterministic analysis of receptor-mediated cell attachment and detachment for a homogeneous cell population. However, initial comparison of a deterministic framework to experimental detachment profiles of model "cells" (antibody-coated latex beads) did not show qualitative or quantitative agreement (Cozens-Roberts, C., D.A. Lauffenburger, and J.A. Quinn. 1990. Biophys. J. 58:857-872). Hence, we determine the contributions of population heterogeneity and probabilistic binding to the detachment behavior of this experimental system which was designed to minimize experimental and theoretical complications. This work also corrects an error in the numerical solution of the probabilistic model of receptor-mediated cell attachment and detachment developed previously (Cozens-Roberts, C., D.A. Lauffenburger, and J.A. Quinn. 1990. Biophys J. 58:841-856). Measurement of the population distribution of the number of receptors per bead has enabled us to explicitly consider the effect of receptor number heterogeneity within the cell-surface contact area. A deterministic framework that incorporates receptor number heterogeneity qualitatively and quantitatively accounts in large part for the model cell detachment data. Using measured and estimated parameter values for the model cell system, we estimate that about 90% of the observed kinetic detachment behavior originates from heterogeneity effects, while about 10% is due to probabilistic binding effects. In general, these relative contributions may differ for other systems.

Measurement of mass transport and reaction parameters in bulk solution using photobleaching. Reaction limited binding regime

Fluorescence recovery after photobleaching (FRAP) has been used previously to investigate the kinetics of binding to biological surfaces. The present study adapts and further develops this technique for the quantification of mass transport and reaction parameters in bulk media. The technique's ability to obtain the bulk diffusion coefficient, concentration of binding sites, and equilibrium binding constant for ligand/receptor interactions in the reaction limited binding regime is assessed using the B72.3/TAG-72 monoclonal antibody/tumor associated antigen interaction as a model in vitro system. Measurements were independently verified using fluorometry. The bulk diffusion coefficient, concentration of binding sites and equilibrium binding constant for the system investigated were 6.1 +/- 1.1 x 10(-7) cm2/s, 4.4 +/- 0.6 x 10(-7) M, and 2.5 +/- 1.6 x 10(7) M-1, respectively. Model robustness and the applicability of the technique for in vivo quantification of mass transport and reaction parameters are addressed. With a suitable animal model, it is believed that this technique is capable of quantifying mass transport and reaction parameters in vivo.

Receptor-mediated cell attachment and detachment kinetics. II. Experimental model studies with the radial-flow detachment assay

Quantitative information regarding the kinetics of receptor-mediated cell adhesion to a ligand-coated surface are crucial for understanding the role of certain key parameters in many physiological and biotechnology-related processes. Here, we use the probabilistic attachment and detachment models developed in the preceding paper to interpret transient data from well-defined experiments. These data are obtained with a simple model cell system that consists of receptor-coated latex beads (prototype cells) and a Radial-Flow Detachment Assay (RFDA) using a ligand-coated glass disc. The receptors and ligands used in this work are complementary antibodies. The beads enable us to examine transient behavior with particles that possess fairly uniform properties that can be varied systematically, and the RFDA is designed for direct observation of adhesion to the ligand-coated glass surface over a range of shear stresses. Our experiments focus on the effects of surface shear stress, receptor density, and ligand density. These data provide a crucial test of the probabilistic framework. We show that these data can be explained with the probabilistic analyses, whereas they cannot be readily interpreted on the basis of a deterministic analysis. In addition, we examine transient data on cell adhesion reported from other assays, demonstrating the consistency of these data with the predictions of the probabilistic models.

Receptor-mediated adhesion phenomena. Model studies with the Radical-Flow Detachment Assay

Receptor-mediated cell adhesion phenomena play a vital role in many physiological and biotechnology-related processes. To investigate the physical and chemical factors that influence the cell/surface interaction, we have used a radial flow device, a so-called Radial-Flow Detachment Assay (RFDA). The RFDA allows us to make direct observations of the detachment process under specified experimental conditions. In results reported here, we have studied the detachment of receptor-coated latex beads (prototype cells) from ligand-coated glass surfaces. The receptors and ligands used in this work are complementary antibodies. The beads enable us to examine several aspects of the adhesion process with particles having uniform properties that can be varied systematically. Advantages of the RFDA are many, especially direct observation of cell detachment over a range of shear stresses with quantitative measurement of the adhesive force. We focus our studies on the effects of ligand and receptor densities, along with the influence of pH and ionic strength of the medium. These data are analyzed with a mathematical model based on the theoretical framework of Bell, G. I. (1978. Science [Wash. DC]. 200:618-627) and Hammer, D. A. and D. A. Lauffenburger (1987. Biophys. J. 52:475-487). We demonstrate experimental validation of a theoretical expression for the critical shear stress for particle detachment, and show that it is consistent with reasonable estimates for the receptor-ligand bond affinity.

Verification of immune response optimality through cybernetic modeling

An immune response cascade that is T cell independent begins with the stimulation of virgin lymphocytes by antigen to differentiate into large lymphocytes. These immune cells can either replicate themselves or differentiate into plasma cells or memory cells. Plasma cells produce antibody at a specific rate up to two orders of magnitude greater than large lymphocytes. However, plasma cells have short life-spans and cannot replicate. Memory cells produce only surface antibody, but in the event of a subsequent infection by the same antigen, memory cells revert rapidly to large lymphocytes. Immunologic memory is maintained throughout the organism's lifetime. Many immunologists believe that the optimal response strategy calls for large lymphocytes to replicate first, then differentiate into plasma cells and when the antigen has been nearly eliminated, they form memory cells. A mathematical model incorporating the concept of cybernetics has been developed to study the optimality of the immune response. Derived from the matching law of microeconomics, cybernetic variables control the allocation of large lymphocytes to maximize the instantaneous antibody production rate at any time during the response in order to most efficiently inactivate the antigen. A mouse is selected as the model organism and bacteria as the replicating antigen. In addition to verifying the optimal switching strategy, results showing how the immune response is affected by antigen growth rate, initial antigen concentration, and the number of antibodies required to eliminate an antigen are included.

Modeling adaptive biological systems

During the evolution of many systems found in nature, both the system composition and the interactions between components will vary. Equating the dimension with the number of different components, a system which adds or deletes components belongs to a class of dynamical systems with a finite dimensional phase space of variable dimension. We present two models of biochemical systems with a variable phase space, a model of autocatalytic reaction networks in the prebiotic soup and a model of the idiotypic network of the immune system. Each model contains characteristic meta-dynamical rules for constructing equations of motion from component properties. The simulation of each model occurs on two levels. On one level, the equations of motion are integrated to determine the state of each component. On a second level, algorithms which approximate physical processes in the real system are employed to change the equations of motion. Models with meta-dynamical rules possess several advantages for the study of evolving systems. First, there are no explicit fitness functions to determine how the components of the model rank in terms of survivability. The success of any component is a function of its relationship to the rest of the system. A second advantage is that since the phase space representation of the system is always finite but continually changing, we can explore a potentially infinite phase space which would otherwise be inaccessible with finite computer resources. Third, the enlarged capacity of systems with meta-dynamics for variation allows us to conduct true evolution experiments. The modeling methods presented here can be applied to many real biological systems. In the two studies we present, we are investigating two apparent properties of adaptive networks. With the simulation of the prebiotic soup, we are most interested in how a chemical reaction network might emerge from an initial state of relative disorder. With the study of the immune system, we study the self-regulation of the network including its ability to distinguish between species which are part of the network and those which are not.

Kinetics of antigen-antibody reactions at solid-liquid interfaces

The kinetics of antigen-antibody reactions is reviewed with special attention paid to the specific properties at solid-liquid interfaces. Theories of possible diffusion limitation in forward reaction rates are compared to experiments. It is found that the intrinsic forward reaction rate in the bimolecular antigen-antibody reaction is normally not limited by diffusion either in solution or at the solid-liquid interface. However, reactions at the solid-liquid interface can be diffusion limited due to depletion of reactants close to the surface. This effect depends on geometry, intrinsic reaction rate and surface concentration of receptor molecules. Normally cell surface reactions are not diffusion limited whereas reactions at artificial surfaces often are limited by diffusion. When not limited by diffusion it is also found that the intrinsic forward and reverse reaction rates are lower for surface reactions compared to reactions in solution. Antigen-antibody reactions at solid-liquid interfaces can often be considered as practically irreversible and limited by mass transport or steric interactions.

External diffusion in solid-phase immunoassays

Calculations are presented describing the influence of external diffusion in the kinetics of solid-phase immunoassays. The analysis is concerned with systems where one reactant is immobilized at the surface of a sphere of arbitrary radius. The solution for a plane surface is found as a limiting case. The factors determining whether the reaction is diffusion or reaction controlled are found to be sphere radius, surface concentration of binding sites, forward reaction rate and diffusion constant of reacting species. Means of determining whether the reaction is diffusion or reaction controlled from observable quantities are described. When applied to heterogeneous antibody-antigen binding it is found that normally the binding to cell-size spheres is not limited by external diffusion. However, when applied to solid-phase assays with high surface concentrations of binding sites immobilized at plane surfaces or macroscopic spheres the binding is found to be diffusion limited. The importance of a mass transfer analysis in this case is also discussed.

A theory of measurement error and its implications for spatial and temporal gradient sensing during chemotaxis

In order that cells respond to environmental cues, they must be able to measure ambient ligand concentration. Concentrations fluctuate, however, because of thermal noise, and one can readily show that estimates based on concentration values at a particular moment will be subject to substantial error. Cells are therefore expected to average their estimates over some limited time period. In this paper we assume that a cell uses fractional receptor occupancy as a measure of ambient ligand concentration and develop general expressions for the error a cell makes because the length of the averaging period is necessarily limited. Our analysis is general, relieving many of the assumptions underlying the seminal work of Berg and Purcell. The most important formal difference is our inclusion of occupancy-dependent dissociation--a phenomenon that has been well-documented for many systems. In addition, our formulation permits signal averaging to begin before chemical equilibrium has been established and it allows binding kinetics to be nonlinear (i.e., biomolecular rather than pseudo-first-order). The results are applied to spatial and temporal concentration gradients. In particular we estimate the minimum averaging times required for cells to detect such gradients under typical in vitro conditions. These estimates involve assigning numerical values to receptor ligand rate constants. If the rate constants are at their maximum possible values (limited only by center of mass diffusion), then either temporal or spatial gradients can be detected in minutes or less. If, however, as suggested by experiments, the rate constants are several orders of magnitude below their diffusion-limited values, then under typical constant gradient conditions the time required to detect a spatial gradient is prohibitively long, whereas temporal gradients can still be detected in reasonable lengths of time. This result was obtained for large cells such as lymphocytes, as well as for the smaller, bacterial cells. The ratio of averaging times for the two mechanisms--amounting to several orders of magnitude--is well beyond what could be reconciled by limitations of the calculation, and strongly suggests heavy reliance on temporal sensing mechanisms under typical in vitro conditions with constant spatial gradients.

Effect of nonspecific forces and finite receptor number on rate constants of ligand--cell bound-receptor interactions

We develop a theory of ligand diffusion in the presence of a central potential toward or away from receptor patches that are uniformly distributed over a spherical cell. The current onto the receptors is reduced to less than that onto the sphere by a factor that is a nonlinear function of the number of free receptors, their size, and their potential energy. Similarly, under conditions defined by the theory, the dissociation rate from a receptor is reduced by the probability of rebinding to some other receptor on the same cell. This complicates the kinetic analysis, leading to the possibility of an occupancy-dependent dissociation rate, but has no effect on the interpretation of thermodynamic data.