Rigorous Mathematical Analysis of the Quasispecies Model: From Manfred Eigen to the Recent Developments

Bratus, Alexander S.; Novozhilov, Artem S.; Semenov, Yuri S.

doi:10.1007/978-3-030-15715-9_2

Cited by 7 publications

(6 citation statements)

References 42 publications

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“…Suppose also that the growth saturation function ϕ is twice differentiable at the point s * = n i=1 u * i . Consider the Jacobian matrix function DF determined by (13). At the steady state u * , its elements can be represented as…”

Section: Steady-state Analysismentioning

confidence: 99%

“…A great deal of mathematical investigation was devoted to study exact properties of the quasispecies and error threshold; see, e. g., the review in [13]. In a nutshell, the exact details of the structure of the mutation-selection equilibrium and the precise position (if it exists at all) of the error threshold depend in a subtle way on the implemented fitness landscape.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Open Quasispecies Models: Stability, Optimization, and Distributed Extension

Yegorov¹,

Novozhilov²,

Bratus³

2019

Preprint

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We suggest a natural approach that leads to a modification of classical quasispecies models and incorporates the possibility of population extinction in addition to growth. The resulting modified models are called open. Their essential properties, regarding in particular equilibrium behavior, are investigated both analytically and numerically. The hallmarks of the quasispecies dynamics, viz. the heterogeneous quasispecies distribution itself and the error threshold phenomenon, can be observed in our models, along with extinction. In order to demonstrate the flexibility of the introduced framework, we study the inverse problem of fitness allocation under the biologically motivated criterion of steady-state fitness maximization. Having in mind the complexity of numerical investigation of high-dimensional quasispecies problems and the fact that the actual number of genotypes or alleles involved in a studied process can be extremely large, we also build continuous-time distributed open quasispecies models. The obtained results may serve as an initial step to developing mathematical models that involve directed therapy against various pathogens.

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Section: Steady-state Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Open Quasispecies Models: Stability, Optimization, and Distributed Extension

Yegorov¹,

Novozhilov²,

Bratus³

2019

Preprint

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“…Additionally, it was found that different viruses within a quasispecies can exhibit a wide range of infectiousness, virulence, and replicative fitness [6,14,15]. Complementary to these developments, the theoretical foundations of quasispecies, proposed originally by Eigen, have been the subject of extensive study in mathematical biology and physics, leading to exact solution methods and applications ranging from B-cell receptor diversity to intra-tumor population dynamics [16][17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%

Modeling the dynamics of within-host viral infection and evolution predicts quasispecies distributions and phase boundaries separating distinct classes of infections

Lewis

Marshall

Jones

2021

Preprint

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We use computational modeling to study within-host viral infection and evolution. In our model, viruses exhibit variable binding to cells, with better infection and replication countered by a stronger immune response and a high rate of mutation. By varying host conditions (permissivity to viral entry T and immune clearance intensity A) for large numbers of cells and viruses, we study the dynamics of how viral populations evolve from initial infection to steady state and obtain a phase diagram of the range of cell and viral responses. We find three distinct replicative strategies corresponding to three physiological classes of viral infections: acute, chronic, and opportunistic. We show similarities between our findings and the behavior of real viral infections such as common flu, hepatitis, and SARS-CoV-2019. The phases associated with the three strategies are separated by a phase transition of primarily first order, in addition to a crossover region. Our simulations also reveal a wide range of physical phenomena, including metastable states, periodicity, and glassy dynamics. Lastly, our results suggest that the resolution of acute viral disease in patients whose immunity cannot be boosted can only be achieved by significant inhibition of viral infection and replication.

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“…These models can be mathematically tractable, allowing for analysis of equilibria and stability in terms of mutation rates and fitness quantities. In particular, the common setting of finite binary sequences, the assumed form of viral genotypes in this current paper, enables geometric or algebraic properties of the binary hypercube space to be exploited for characterizing equilibrium distributions [7]. Inclusion of viral mutation from multiple dynamic immune response populations complicates matters, as neither the virus strain fitness or immune response strength simply determine epitope escape [18,29].…”

Section: Introductionmentioning

confidence: 99%

Virus-immune dynamics determined by prey-predator interaction network and epistasis in viral fitness landscape

Browne¹,

Fadoua²

2021

Preprint

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Population dynamics and evolutionary genetics underly the structure of ecosystems, changing on the same timescale for interacting species with rapid turnover, such as virus (e.g. HIV) and immune response. Thus, an important problem in mathematical modeling is to connect ecology, evolution and genetics, which often have been treated separately. Here, extending analysis of multiple virus and immune response populations in a resource -prey (consumer) -predator model from Browne and Smith [9], we show that long term dynamics of viral mutants evolving resistance at distinct epitopes (viral proteins targeted by immune responses) are governed by epistasis in the virus fitness landscape. In particular, the stability of persistent equilibrium virus-immune (preypredator) network structures, such as nested and one-to-one, and bifurcations are determined by a collection of circuits defined by combinations of viral fitnesses that are minimally additive within a hypercube of binary sequences representing all possible viral epitope sequences ordered according to immunodominance hierarchy. Numerical solutions of our ordinary differential equation system, along with an extended stochastic version including random mutation, demonstrate how pairwise or multiplicative epistatic interactions shape viral evolution against concurrent immune responses and convergence to the multi-variant steady state predicted by theoretical results. Furthermore, simulations illustrate how periodic infusions of subdominant immune responses can induce a bifurcation in the persistent viral strains, offering superior host outcome over an alternative strategy of immunotherapy with strongest immune response.

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Rigorous Mathematical Analysis of the Quasispecies Model: From Manfred Eigen to the Recent Developments

Cited by 7 publications

References 42 publications

Open Quasispecies Models: Stability, Optimization, and Distributed Extension

Open Quasispecies Models: Stability, Optimization, and Distributed Extension

Modeling the dynamics of within-host viral infection and evolution predicts quasispecies distributions and phase boundaries separating distinct classes of infections

Virus-immune dynamics determined by prey-predator interaction network and epistasis in viral fitness landscape

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