Restricted occupancy models for neutralization of HIV virions and populations

Multi-scale modeling of drug binding kinetics to predict drug efficacy

Optimizing drug therapies for any disease requires a solid understanding of pharmacokinetics (the drug concentration at a given time point in different body compartments) and pharmacodynamics (the effect a drug has at a given concentration). Mathematical models are frequently used to infer drug concentrations over time based on infrequent sampling and/or in inaccessible body compartments. Models are also used to translate drug action from in vitro to in vivo conditions or from animal models to human patients. Recently, mathematical models that incorporate drug-target binding and subsequent downstream responses have been shown to advance our understanding and increase predictive power of drug efficacy predictions. We here discuss current approaches of modeling drug binding kinetics that aim at improving model-based drug development in the future. This in turn might aid in reducing the large number of failed clinical trials.

Estimating treatment prolongation for persistent infections

Treatment of infectious diseases is often long and requires patients to take drugs even after they have seemingly recovered. This is because of a phenomenon called persistence, which allows small fractions of the bacterial population to survive treatment despite being genetically susceptible. The surviving subpopulation is often below detection limit and therefore is empirically inaccessible but can cause treatment failure when treatment is terminated prematurely. Mathematical models could aid in predicting bacterial survival and thereby determine sufficient treatment length. However, the mechanisms of persistence are hotly debated, necessitating the development of multiple mechanistic models. Here we develop a generalized mathematical framework that can accommodate various persistence mechanisms from measurable heterogeneities in pathogen populations. It allows the estimation of the relative increase in treatment length necessary to eradicate persisters compared to the majority population. To simplify and generalize, we separate the model into two parts: the distribution of the molecular mechanism of persistence in the bacterial population (e.g. number of efflux pumps or target molecules, growth rates) and the elimination rate of single bacteria as a function of that phenotype. Thereby, we obtain an estimate of the required treatment length for each phenotypic subpopulation depending on its size and susceptibility.

Bio-inspired computational heuristics to study models of HIV infection of CD4+ T-cell

In this work, biologically-inspired computing framework is developed for HIV infection of CD4[Formula: see text] T-cell model using feed-forward artificial neural networks (ANNs), genetic algorithms (GAs), sequential quadratic programming (SQP) and hybrid approach based on GA-SQP. The mathematical model for HIV infection of CD4[Formula: see text] T-cells is represented with the help of initial value problems (IVPs) based on the system of ordinary differential equations (ODEs). The ANN model for the system is constructed by exploiting its strength of universal approximation. An objective function is developed for the system through unsupervised error using ANNs in the mean square sense. Training with weights of ANNs is carried out with GAs for effective global search supported with SQP for efficient local search. The proposed scheme is evaluated on a number of scenarios for the HIV infection model by taking the different levels for infected cells, natural substitution rates of uninfected cells, and virus particles. Comparisons of the approximate solutions are made with results of Adams numerical solver to establish the correctness of the proposed scheme. Accuracy and convergence of the approach are validated through the results of statistical analysis based on the sufficient large number of independent runs.

Predicting HIV-1 transmission and antibody neutralization efficacy in vivo from stoichiometric parameters

The potential of broadly neutralizing antibodies targeting the HIV-1 envelope trimer to prevent HIV-1 transmission has opened new avenues for therapies and vaccines. However, their implementation remains challenging and would profit from a deepened mechanistic understanding of HIV-antibody interactions and the mucosal transmission process. In this study we experimentally determined stoichiometric parameters of the HIV-1 trimer-antibody interaction, confirming that binding of one antibody is sufficient for trimer neutralization. This defines numerical requirements for HIV-1 virion neutralization and thereby enables mathematical modelling of in vitro and in vivo antibody neutralization efficacy. The model we developed accurately predicts antibody efficacy in animal passive immunization studies and provides estimates for protective mucosal antibody concentrations. Furthermore, we derive estimates of the probability for a single virion to start host infection and the risks of male-to-female HIV-1 transmission per sexual intercourse. Our work thereby delivers comprehensive quantitative insights into both the molecular principles governing HIV-antibody interactions and the initial steps of mucosal HIV-1 transmission. These insights, alongside the underlying, adaptable modelling framework presented here, will be valuable for supporting in silico pre-trial planning and post-hoc evaluation of HIV-1 vaccination or antibody treatment trials.

Mathematical Model for an Effective Management of HIV Infection

Human immunodeficiency virus infection destroys the body immune system, increases the risk of certain pathologies, damages body organs such as the brain, kidney, and heart, and causes death. Unfortunately, this infectious disease currently has no cure; however, there are effective retroviral drugs for improving the patients' health conditions but excessive use of these drugs is not without harmful side effects. This study presents a mathematical model with two control variables, where the uninfected CD4⁺T cells follow the logistic growth function and the incidence term is saturated with free virions. We use the efficacy of drug therapies to block the infection of new cells and prevent the production of new free virions. Our aim is to apply optimal control approach to maximize the concentration of uninfected CD4⁺T cells in the body by using minimum drug therapies. We establish the existence of an optimal control pair and use Pontryagin's principle to characterize the optimal levels of the two controls. The resulting optimality system is solved numerically to obtain the optimal control pair. Finally, we discuss the numerical simulation results which confirm the effectiveness of the model.

The HIV-1 Entry Process: A Stoichiometric View

HIV-1 infection starts with fusion of the viral and the host cell membranes, a process mediated by the HIV-1 envelope glycoprotein trimer. The number of trimers required to complete membrane fusion, referred to as HIV-1 entry stoichiometry, remains under debate. A precise definition of HIV-1 entry stoichiometry is important as it reflects the efficacy of the viral entry process and steers the infectivity of HIV-1 virion populations. Initial estimates suggested a unanimous entry stoichiometry across HIV-1 strains while recent findings showed that HIV-1 strains can differ in entry stoichiometry. Here, we review current analyses of HIV-1 entry stoichiometry and point out future research directions to further define the interplay between entry stoichiometry, virus entry fitness, transmission, and susceptibility to antibody neutralization.

Different infectivity of HIV-1 strains is linked to number of envelope trimers required for entry

HIV-1 enters target cells by virtue of envelope glycoprotein trimers that are incorporated at low density in the viral membrane. How many trimers are required to interact with target cell receptors to mediate virus entry, the HIV entry stoichiometry, still awaits clarification. Here, we provide estimates of the HIV entry stoichiometry utilizing a combined approach of experimental analyses and mathematical modeling. We demonstrate that divergent HIV strains differ in their stoichiometry of entry and require between 1 to 7 trimers, with most strains depending on 2 to 3 trimers to complete infection. Envelope modifications that perturb trimer structure lead to an increase in the entry stoichiometry, as did naturally occurring antibody or entry inhibitor escape mutations. Highlighting the physiological relevance of our findings, a high entry stoichiometry correlated with low virus infectivity and slow virus entry kinetics. The entry stoichiometry therefore directly influences HIV transmission, as trimer number requirements will dictate the infectivity of virus populations and efficacy of neutralizing antibodies. Thereby our results render consideration of stoichiometric concepts relevant for developing antibody-based vaccines and therapeutics against HIV.

Our estimates of the HIV-1 entry stoichiometry, that is the number of envelope glycoprotein trimers needed to mediate fusion of viral and target cell membrane, close an important gap in our understanding of the HIV entry process. As we show, stoichiometric requirements for envelope trimers differ between HIV strains and steer virus entry efficacy and virus entry kinetics. Thus, the entry stoichiometry has important implications for HIV transmission, as demands on trimer numbers will dictate the infectivity of virus populations, target cell preferences and virus inactivation by trimer-targeting inhibitors and neutralizing antibodies. Beyond this, our data contribute to the general understanding of mechanisms and energetic requirements of protein-mediated membrane fusion, as HIV entry proved to follow similar stoichiometries as described for Influenza virus HA and SNARE protein mediated membrane fusion. In summary, our findings provide a relevant contribution towards a refined understanding of HIV-1 entry and pathogenesis with particular importance for ongoing efforts to generate neutralizing antibody based therapeutics and vaccines targeting the HIV-1 envelope trimer.

The role of chance in primate lentiviral infectivity: from protomer to host organism

Infection is best described as a stochastic process. Whether a host becomes infected upon exposure has a strong random element. The same applies to cells exposed to virions. In this review, we show how the mathematical formalism for stochastic processes has been used to describe and understand the infection by the Human and Simian Immunodeficiency Virus on different levels. We survey quantitative studies on the establishment of infection in the host (the organismal level) and on the infection of target cells (the cellular and molecular level). We then discuss how a synthesis of the approaches across these levels could give rise to a predictive framework for assessing the efficacy of microbicides and vaccines.

Molecular determinants of the ratio of inert to infectious virus particles

The ratio of virus particles to infectious units is a classic measurement in virology and ranges widely from several million to below 10 for different viruses. Much evidence suggests a distinction be made between infectious and infecting particles or virions: out of many potentially infectious virions, few infect under regular experimental conditions, largely because of diffusion barriers. Still, some virions are inert from the start; others become defective through decay. And with increasing cell- and molecular-biological knowledge of each step in the replicative cycle for different viruses, it emerges that many processes entail considerable losses of potential viral infectivity. Furthermore, all-or-nothing assumptions about virion infectivity are flawed and should be replaced by descriptions that allow for spectra of infectious propensities. A more realistic understanding of the infectivity of individual virions has both practical and theoretical implications for virus neutralization, vaccine research, antiviral therapy, and the use of viral vectors.

Neutralization of Virus Infectivity by Antibodies: Old Problems in New Perspectives

Neutralizing antibodies (NAbs) can be both sufficient and necessary for protection against viral infections, although they sometimes act in concert with cellular immunity. Successful vaccines against viruses induce NAbs but vaccine candidates against some major viral pathogens, including HIV-1, have failed to induce potent and effective such responses. Theories of how antibodies neutralize virus infectivity have been formulated and experimentally tested since the 1930s; and controversies about the mechanistic and quantitative bases for neutralization have continually arisen. Soluble versions of native oligomeric viral proteins that mimic the functional targets of neutralizing antibodies now allow the measurement of the relevant affinities of NAbs. Thereby the neutralizing occupancies on virions can be estimated and related to the potency of the NAbs. Furthermore, the kinetics and stoichiometry of NAb binding can be compared with neutralizing efficacy. Recently, the fundamental discovery that the intracellular factor TRIM21 determines the degree of neutralization of adenovirus has provided new mechanistic and quantitative insights. Since TRIM21 resides in the cytoplasm, it would not affect the neutralization of enveloped viruses, but its range of activity against naked viruses will be important to uncover. These developments bring together the old problems of virus neutralization-mechanism, stoichiometry, kinetics, and efficacy-from surprising new angles.

Mathematical model of multivalent virus-antibody complex formation in humans following acute and chronic HIV infections

Antibodies that bind viral surface proteins can limit the spread of the infection through neutralizing and non-neutralizing functions. During both acute and chronic Human Immunodeficiency Virus infection, antibody-virion immune complexes are formed, but fail to ensure protection. In this study, we develop a mathematical model of multivalent antibody binding and use it to determine the dynamical interactions that lead to immune complexes formation and the role of complexes with increased numbers of bound antibodies in the pathogenesis of the disease. We compare our predictions with published temporal virus and immune complex data from acute infected patients. Finally, we derive quantitative and qualitative conditions needed for antibody-induced protection.

Antibody responses during hepatitis B viral infection

Hepatitis B is a DNA virus that infects liver cells and can cause both acute and chronic disease. It is believed that both viral and host factors are responsible for determining whether the infection is cleared or becomes chronic. Here we investigate the mechanism of protection by developing a mathematical model of the antibody response following hepatitis B virus (HBV) infection. We fitted the model to data from seven infected adults identified during acute infection and determined the ability of the virus to escape neutralization through overproduction of non-infectious subviral particles, which have HBs proteins on their surface, but do not contain nucleocapsid protein and viral nucleic acids. We showed that viral clearance can be achieved for high anti-HBV antibody levels, as in vaccinated individuals, when: (1) the rate of synthesis of hepatitis B subviral particles is slow; (2) the rate of synthesis of hepatitis B subviral particles is high but either anti-HBV antibody production is fast, the antibody affinity is high, or the levels of pre-existent HBV-specific antibody at the time of infection are high, as could be attained by vaccination. We further showed that viral clearance can be achieved for low equilibrium anti-HBV antibody levels, as in unvaccinated individuals, when a strong cellular immune response controls early infection.

Differential binding of neutralizing and non-neutralizing antibodies to native-like soluble HIV-1 Env trimers, uncleaved Env proteins, and monomeric subunits

Background

The trimeric envelope glycoproteins (Env) on the surface of HIV-1 virions are the targets for neutralizing antibodies (NAbs). No candidate HIV-1 immunogen has yet induced potent, broadly active NAbs (bNAbs). Part of the explanation may be that previously tested Env proteins inadequately mimic the functional, native Env complex. Trimerization and the proteolytic processing of Env precursors into gp120 and gp41 profoundly alter antigenicity, but soluble cleaved trimers are too unstable to serve as immunogens. By introducing stabilizing mutations (SOSIP), we constructed soluble, cleaved Env trimers derived from the HIV-1 subtype A isolate BG505 that resemble native Env spikes on virions both structurally and antigenically.

Results

We used surface plasmon resonance (SPR) to quantify antibody binding to different forms of BG505 Env: the proteolytically cleaved SOSIP.664 trimers, cleaved gp120-gp41_ECTO protomers, and gp120 monomers. Non-NAbs to the CD4-binding site bound only marginally to the trimers but equally well to gp120-gp41_ECTO protomers and gp120 monomers, whereas the bNAb VRC01, directed to the CD4bs, bound to all three forms. In contrast, bNAbs to V1V2 glycan-dependent epitopes bound preferentially (PG9 and PG16) or exclusively (PGT145) to trimers. We also explored the antigenic consequences of three different features of SOSIP.664 gp140 trimers: the engineered inter-subunit disulfide bond, the trimer-stabilizing I559P change in gp41_ECTO, and proteolytic cleavage at the gp120-gp41_ECTO junction. Each of these three features incrementally promoted native-like trimer antigenicity. We compared Fab and IgG versions of bNAbs and validated a bivalent model of IgG binding. The NAbs showed widely divergent binding kinetics and degrees of binding to native-like BG505 SOSIP.664. High off-rate constants and low stoichiometric estimates of NAb binding were associated with large amounts of residual infectivity after NAb neutralization of the corresponding BG505.T332N pseudovirus.

Conclusions

The antigenicity and structural integrity of cleaved BG505 SOSIP.664 trimers render these proteins good mimics of functional Env spikes on virions. In contrast, uncleaved gp140s antigenically resemble individual gp120-gp41_ECTO protomers and gp120 monomers, but not native trimers. Although NAb binding to functional trimers may thus be both necessary and sufficient for neutralization, the kinetics and stoichiometry of the interaction influence the neutralizing efficacy of individual NAbs.

Virus neutralisation: new insights from kinetic neutralisation curves

Antibodies binding to the surface of virions can lead to virus neutralisation. Different theories have been proposed to determine the number of antibodies that must bind to a virion for neutralisation. Early models are based on chemical binding kinetics. Applying these models lead to very low estimates of the number of antibodies needed for neutralisation. In contrast, according to the more conceptual approach of stoichiometries in virology a much higher number of antibodies is required for virus neutralisation by antibodies. Here, we combine chemical binding kinetics with (virological) stoichiometries to better explain virus neutralisation by antibody binding. This framework is in agreement with published data on the neutralisation of the human immunodeficiency virus. Knowing antibody reaction constants, our model allows us to estimate stoichiometrical parameters from kinetic neutralisation curves. In addition, we can identify important parameters that will make further analysis of kinetic neutralisation curves more valuable in the context of estimating stoichiometries. Our model gives a more subtle explanation of kinetic neutralisation curves in terms of single-hit and multi-hit kinetics.

How many antibodies have to bind to a virus particle such that it is prevented from infecting a cell? This seemingly simple question has not been answered yet. However, this number is crucial to determine whether a vaccine can stimulate the immune system to elicit enough antibodies to neutralise virus before starting an infection. Two different approaches have been applied to answer this question, leading to contradictory results. One approach is inspired by concepts from binding kinetics, the other approach is a more conceptual one. Here, I describe the advantages and disadvantages of either approaches and condense the advantages of both into one model framework. I show under which conditions the framework can be used to identify the number of neutralising antibodies. In addition, this model can explain why viruses might not completely loose their infection potential even when there is a huge excess of antibodies.

Analysis of the subunit stoichiometries in viral entry

Virions of the Human Immunodeficiency Virus (HIV) infect cells by first attaching with their surface spikes to the CD4 receptor on target cells. This leads to conformational changes in the viral spikes, enabling the virus to engage a coreceptor, commonly CCR5 or CXCR4, and consecutively to insert the fusion peptide into the cellular membrane. Finally, the viral and the cellular membranes fuse. The HIV spike is a trimer consisting of three identical heterodimers composed of the gp120 and gp41 envelope proteins. Each of the gp120 proteins in the trimer is capable of attaching to the CD4 receptor and the coreceptor, and each of the three gp41 units harbors a fusion domain. It is still under debate how many of the envelope subunits within a given trimer have to bind to the CD4 receptors and to the coreceptors, and how many gp41 protein fusion domains are required for fusion. These numbers are referred to as subunit stoichiometries. We present a mathematical framework for estimating these parameters individually by analyzing infectivity assays with pseudotyped viruses. We find that the number of spikes that are engaged in mediating cell entry and the distribution of the spike number play important roles for the estimation of the subunit stoichiometries. Our model framework also shows why it is important to subdivide the question of the number of functional subunits within one trimer into the three different subunit stoichiometries. In a second step, we extend our models to study whether the subunits within one trimer cooperate during receptor binding and fusion. As an example for how our models can be applied, we reanalyze a data set on subunit stoichiometries. We find that two envelope proteins have to engage with CD4-receptors and coreceptors and that two fusion proteins must be revealed within one trimer for viral entry. Our study is motivated by the mechanism of HIV entry but the experimental technique and the model framework can be extended to other viral systems as well.

How do antigenically varying pathogens avoid cross-reactive responses to invariant antigens?

Pathogens such as trypanosomes and malaria use antigenic variation to evade immune responses and prolong the duration of infections. As pathogens typically express more than one antigen, even relatively rare conserved antigens might be expected to trigger cross-reactive immune responses capable of clearing the infection. We use simple mathematical models that explicitly consider the dynamic interplay between the replicating pathogen, immune responses to different antigens and immune exhaustion to explore how pathogens can escape the responses to both variable and invariant (conserved) antigens. Our results suggest two hypotheses. In the first, limited quantities of invariant antigens on each pathogen may lead to saturation in killing by cross-reactive responses. In the second, antigenic variation of the dominant antigens prolongs the duration of infection sufficiently to allow for exhaustion of the cross-reactive responses to subdominant, invariant epitopes prior to their being able to control the infection. These hypotheses make distinct predictions: the former predicts that cross-reactive responses will always be ineffective while the latter predicts that appropriately timed treatment could, by preventing exhaustion, lead to the generation of long-lasting protective cross-reactive immunity and thus act similarly to a vaccine.