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A Mathematical Model of Continuous HIV Mutations Eluding Immune Defence

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A Mathematical Model of Continuous HIV Mutations Eluding Immune Defence Journal of Theoretical Medicine, 2002 Vol. 4 (4), pp. 241–249 A Mathematical Model of Continuous HIV Mutations Eluding Immune Defence D.P. WILSON* and D.L.S. MCELWAIN Centre in Statistical Science and Industrial Mathematics, Queensland University of Technology, Brisbane, Australia (Received 2 July 2002; In final form 9 December 2002) We develop a description of HIV mutations based upon a continuum representation of the fitness of the virus, including the interaction of the virus with both specific Th1 lymphocytes as well as cross-reactive cells. This deterministic model allows a straightforward measure of the diversity of viral population and reproduces the observed increase in diversity as the disease progresses in an untreated patient. We use the diversity threshold theory, extending the modelling to track mutations on a continuum. When the diversity threshold is exceeded, the host immune system collapses. Keywords: HIV; Mutation; Diversity Threshold; Mathematical Model INTRODUCTION cellular DNA, translation forming polypeptides and protease, and cleaving of polypeptides by protease into The development of AIDS is associated with the depletion functional HIV proteins, the infected cell is stimulated to of the most crucial cell type of the immune system, the produce copies of the virus, releasing hundreds, or even CD4þ helper T lymphocyte. This occurs after a period of thousands, of free virus particles that can infect other clinical latency varying in length from 1–2 years to more CD4þ T-cells. We assume that peripheral blood CD4 than 15 years. The human immunodeficiency virus (HIV) counts (generally 1000 per ml in healthy persons) are a and the immune system interact dynamically over the good indicator of CD4 densities in the body. Since the entire course of infection. HIV replicates in large numbers CD4þ cells are depleted over time, strengthening each day and also contributes to the destruction of cells of cytotoxic responses cannot occur. The loss of immune the immune system. This growth is usually balanced by a competence enables normally controllable diseases to strong defensive response which keeps the virus in check. cause opportunistic infections. In addition to a reduced However, eventually the balance of power is transferred so CD4 count, one of the reasons for the specific immune that HIV gains the upper hand and causes this severe response dysfunction is that viral epitopes also change damage to the immune system. This defines the period of resulting in escape from immunological response. disease known as AIDS. Initially, the transformation of immune-sensitive to As with all viruses, HIV cannot reproduce by itself. resistant genotypes occurs by the generation of mutations It relies on a host to assist reproduction. HIV targets CD4þ primarily due to reverse transcriptase errors. HIV’s RNA T cells which are the most abundant white blood cells of genome is reverse transcribed into DNA when the virus the immune system. Since CD4þ T cells play a key role in replicates. Amplification of mutant genotypes by selec- orchestrating both the humoral and the cellular immune tion, and then fixation of the resistant population aids in response, the infection and destruction of this cell type by the problem of resistance. These mutations are changes in HIV has a devastating impact on the immune response. DNA structure, and, therefore, changes in protein and When HIV enters the body, it targets cells with CD4þ phenotype. The mutations we consider are caused by receptors. The gp120 protein on the viral particle binds to spontaneous errors during DNA replication (base substi- the CD4þ T cell receptors and the HIV particle injects its tutions), or frame shift mutation (addition or deletion of a core. HIV, being a retrovirus carries a copy of its RNA, base). They are heritable and may be beneficial, neutral or which is first transcribed into DNA. After the processes of detrimental to an organism. In contrast to most other reverse transcription, integration of viral DNA into infectious agents, HIV combines a very high mutation rate *Corresponding author. E-mail: [email protected] ISSN 1027-3662 print/ISSN 1607-8578 online q 2002 Taylor & Francis Ltd DOI: 10.1080/1027366031000088438 242 D.P. WILSON AND D.L.S. MCELWAIN with a very large production of virions. The extreme diversification results. This antigenic diversity makes it heterogeneity and diversity of HIV makes the design difficult for the immune system to control the different of effective vaccines extremely difficult (Domingo et al., mutants simultaneously and the virus runs ahead of the 1997). A virus’ biological fitness determines its survival. immune response. This is also because there is an It is this phenomenon, of viral mutation, causing asymmetric interaction between immunological specificity escape from immunological attack, that we address in and viral diversity. Martin Nowak was the first to point out this paper. this asymmetry with the diversity threshold model (See The understanding of the dynamics of antigenic escape Nowak, 1990; 1992; Nowak et al., 1991; Nowak and May, from immunological response has been that a mutation 1992; 1993). That is, each viral strain can infect all immune may enable the virus to have a selective advantage. cells with CD4þ protein, but individual strain-specific Mansky and Temin (1995) found that the level of genetic immune cells can only attack specific virus strains. The variation of HIV is high relative to similar retroviruses, outcome of mutation is that each viral genome must be although not as high as earlier predicted. They were able viewed as being different from any other but at the same to estimate the mutation rate for HIV (see below). time viral strains may be thought of as populations of There is measurable variation among the replication closely related genomes. The immune system is then rates of HIV strains in CD4þ cells (Nowak et al., 1991). persistently confronted with new antigen targets before it Mathematical models for the interaction of HIV and has built up a defence to antigen already present. CD4þ suggest that evolutionary forces may drive It has also been suggested that viral genetic diversity is selection for more virulent strains (May et al., 1990; caused by the host immune response. Findings by DeBoer and Boerlijst, 1994; Maree et al., 2000). The Lukashov et al. (1995) and Wolinsky et al. (1996) observations that the apparent diversity of viral strains indicate that HIV undergoes adaptive evolution in vivo, increases with time (Balfe et al., 1990; Holmes et al., in response to selective pressure exerted by the host 1992) has eventually lead to the conclusion that immune system. Evidence for immune-response depen- evolutionary dynamics of HIV quasispecies is not a dence of virus mutation can be observed by noting that consequence but a cause of immunodeficiency (Nowak viral genetic diversity is greatest in clinically healthy and McMichael, 1995). In this paper, we suggest a novel individuals and much less in persons with AIDS, where approach to considering viral diversity and the progression the immune response has become severely impaired. of mutation towards diversity threshold at which time Many have mathematically modelled virus mutation in AIDS results. a discrete form in an attempt to understand this The number of immunologically different mutants phenomenon (see, for example, Colgrove and Japour, appears to be very large. Genetic variation is not uniform 1991; DeBoer and Boerlijst, 1994; Kirschner and Webb, throughout the HIV genome. There is more variability in 1997). But since there are a large number of mutations and the env gene than in the gag or pol genes. Within the env each is considered to be very closely related to its parent gene there are five hypervariable regions. Much attention strain, it seems that a reasonable proposal is to model has been paid to the third hypervariable region, the V3 mutation in a continuous form. This gives rise to this novel loop because it has immunodominant properties. The continuous model. The model of Nowak et al. (1991) combinatorial possibilities of the 19 variable amino acids involving the tracking of many viral variants is extended in the V3 loop of the env protein is 1920, and furthermore here whereby antigenic diversity and immune response the shape of the loop can also be altered by mutations in are considered as a continuum. In our model, we assume other parts of the envelope protein (Nowak et al., 1991). that all possible mutants will exist at all times, even if at For information on genetic and antigenic variation of the minute concentrations, and the viral distribution will V3 loop refer to (Simmonds et al., 1990; Moore and Nara, change form or shift as mutation occurs. 1991; Wolfs et al., 1991). The complexity of the retroviral lifecycle results in difficulty in estimating the rate of mutation. However, MODELLING VIRAL MUTATION Mansky and Temin (1995) state that the error rate has been estimated as 3.4 £ 1025 per base per replication cycle. In this section, we present the essential mutation concepts Since the average HIV genome has approximately 104 for our mathematical model that explores antigenic drift of bases, the average changes per genome is approximately HIV. Mutations are assumed to involve small changes in 0.34 per replication cycle (Colgrove and Japour, 1991). the gene sequence of a viral strain. These small sequence In addition to the occurrence of large antigenic variation changes may alter the viral properties considerably. that occurs due to this probability, a very large number of However, we consider viruses associated with a virions are produced each day during the asymptomatic neighbourhood of sequences in sequence space around a period of HIV progression before AIDS (Perelson and given sequence to have very closely related properties. Nelson, 1999). Here, we are assuming the existence of criteria for During early stages of infection the viral strains that are defining a neighbourhood of gene sequences in sequence most prevalent in initial infection dominate the dynamics.
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