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Probability of viral entry as a function of receptor concentration c ∞ , and viral degradation µ 0 , for various D, when the number of receptors needed for fusion is N ) 4 (left) and N ) 8 (right). Here, we assume random receptor binding and let degradation act only at the n ) 0 state, with magnitude µ 0 . Other parameters used are κ in ) q ) 0.0001. All quantities are dimensionless.
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Enveloped viruses attach to host cells by binding to receptors on the cell surface. For many viruses, entry occurs via membrane fusion after a sufficient number of receptors have engaged ligand proteins on the virion. Under conditions where the cell surface receptor densities are low, recruitment of receptors may be limited by diffusion rather than...
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... Receptor Binding. In this section, we study the case of random receptor binding, where R ) 1. We first assume that cellular degradation acts only on the "naked" glycoprotein spike so that the nondimensional degradation rates µ 0 > 0 and µ n ) 0 for n * 0. Figure 2 shows the probability P in of the virus penetrating the cell as a function of the nondimen- sional receptor concentration c ∞ , degradation rate µ 0 , and receptor diffusion coefficient D, when the number of receptors necessary for entry is N ) 4 (left) and N ) 8 (right). Nondimensional kinetic rates are set to κ in ) q ) 0.0001. ...
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... shown in Figure 2, when µ 0 ) 0, the time-integrated entry probability P in ) 1 for all positive receptor diffusion coefficients and receptor concentrations. The fact that P in is a decreasing function of µ 0 is expected since only degradation represented by µ 0 > 0 can diminish the total entry probability. ...
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... fact that P in is a decreasing function of µ 0 is expected since only degradation represented by µ 0 > 0 can diminish the total entry probability. Even in the presence of degradation (µ 0 > 0), the fusion probability P in f 1 provided the receptor concentration c ∞ f ∞. Figure 2 also confirms that P in is an increasing function of the receptor diffusion coefficient D. Indeed, increasing D replenishes the concentration of receptors near the glycoprotein spike, speeding the receptor engagement process and increasing the probability of fusion. These qualitative results reveal that as long as c(r ) 1,t) is sufficiently large, either due to large D or large c ∞ , P in will be near unity. ...
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... the range of the parameters c ∞ , µ 0 , and D shown in Figure 2, the entry probabilities remain relatively insensitive to changes in the number of receptors N necessary for fusion when N goes from N ) 4 to N ) 8. However, there is an interesting quantitative transition in P in . ...
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... Figure 4, we plot the conditional mean time to viral entry, T (using eq 10), as a function of receptor concentration c ∞ and degradation rate µ 0 . We used the same parameters values as those used to generate Figure 2. Mean entry times are rather insensitive to receptor diffusivity D except at very low receptor concentrations c ∞ where receptor binding is highly diffusion- limited. ...
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... increasing the diffusion coefficient does not significantly increase the entry probability P in , the entry behavior in this regime is dominated by the degradation term. However, compared to the case where degradation occurs only at the receptor-free state (µ n ) 0 for n * 0, as shown in Figure 2), uniform degradation dramatically decreases the entry prob- ability as µ n ) µ 0 /(N + 1) increases. Even though the relative total strength of degradation is the same as that in Figure 2, the virus can now be degraded at any state n before the final irreversible fusion process, and it is not enough to escape the n ) 0 state to avoid degradation. ...
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... compared to the case where degradation occurs only at the receptor-free state (µ n ) 0 for n * 0, as shown in Figure 2), uniform degradation dramatically decreases the entry prob- ability as µ n ) µ 0 /(N + 1) increases. Even though the relative total strength of degradation is the same as that in Figure 2, the virus can now be degraded at any state n before the final irreversible fusion process, and it is not enough to escape the n ) 0 state to avoid degradation. P in is thus lower in Figure 5 than in Figure 2. ...
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... though the relative total strength of degradation is the same as that in Figure 2, the virus can now be degraded at any state n before the final irreversible fusion process, and it is not enough to escape the n ) 0 state to avoid degradation. P in is thus lower in Figure 5 than in Figure 2. ...
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... can also study how the relative magnitude of the kinetic rate constants affects the viral entry probabilities. Upon varying the unbinding rate constant q, and letting degradation act only on the n ) 0 unbound state, the probabilities P in are monotonic in D and c ∞ and quite similar to those shown in Figure 2 for each value of q. However, nonmonotonic behavior emerges in the dependence of the conditional mean entry time T on c ∞ and q, as shown in Figure 7 for N ) 4 (left) and N ) 8 (right). ...
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... Receptor Binding. In this section, we consider viral entry dynamics for the R ) 0 case of sequential adsorption using the same parameter values used in the previous sections and in Figures 2 and 4. Here, we only consider the case when degradation acts only on the n ) 0 state, with magnitude µ 0 . ...
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Citations
... Regarding the simulation of the viral entry into a cell membrane via endocytosis (Fig. 5), such important aspects were involved as the contact mechanics of viruses onto a flexible membrane [30, 82, 83 ,84, 85], effects of virus size and ligand density on the membrane wrapping [86], as well as, influence of receptor concentration gradient on the ligand-receptor binding [87]. The change of receptor density over the host membrane was described [37,88,89] by the Fick's second law [90,91,92], however, without consideration of diffusion induced stresses [93,94,95,96,97,98]. Continuum modeling was used for adhesive contact between the virus and cell membrane, driven by adhesion [82,83,84] or driven by adhesion and, additionally, by external displacement (or force) [30,85], however, under the assumption of linear elasticity. ...
Introduction. Viruses are a large group of pathogens that have been identified to infect animals, plants, bacteria and even other viruses. The 2019 novel coronavirus SARS-CoV-2 remains a constant threat to the human population. Viruses are biological objects with nanometric dimensions (typically from a few tens to several hundreds of nanometers). They are considered as the biomolecular substances composed of genetic materials (RNA or DNA), protecting capsid proteins and sometimes also of envelopes.
Objectives. The goal of the present review is to help predict the response and even destructuration of viruses taking into account the influence of different environmental factors, such as, mechanical loads, thermal changes, electromagnetic field, chemical changes and receptor binding on the host membrane. These environmental factors have significant impact on the virus.
Materials and methods. The study of viruses and virus-like structures has been analyzed using models and methods of nonlinear mechanics. In this regard, quantum, molecular and continuum descriptions in virus mechanics have been considered. Application of single molecule manipulation techniques, such as, atomic force microcopy, optical tweezers and magnetic tweezers has been discussed for a determination of the mechanical properties of viruses. Particular attention has been given to continuum damage–healing mechanics of viruses, proteins and virus-like structures. Also, constitutive modeling of viruses at large strains is presented. Nonlinear elasticity, plastic deformation, creep behavior, environmentally induced swelling (or shrinkage) and piezoelectric response of viruses were taken into account. Integrating a constitutive framework into ABAQUS, ANSYS and in-house developed software has been discussed.
Conclusion. Link between virus structure, environment, infectivity and virus mechanics may be useful to predict the response and destructuration of viruses taking into account the influence of different environmental factors. Computational analysis using such link may be helpful to give a clear understanding of how neutralizing antibodies and T cells interact with the 2019 novel coronavirus SARS-CoV-2.
... Regarding the simulation of the viral entry into a cell membrane via endocytosis (Fig. 5), such important aspects were involved as the contact mechanics of viruses onto a flexible membrane [30, 82, 83 ,84, 85], effects of virus size and ligand density on the membrane wrapping [86], as well as, influence of receptor concentration gradient on the ligand-receptor binding [87]. The change of receptor density over the host membrane was described [37,88,89] by the Fick's second law [90,91,92], however, without consideration of diffusion induced stresses [93,94,95,96,97,98]. Continuum modeling was used for adhesive contact between the virus and cell membrane, driven by adhesion [82,83,84] or driven by adhesion and, additionally, by external displacement (or force) [30,85], however, under the assumption of linear elasticity. ...
Introdution. Viruses are a large group of pathogens that have been identified to infect animals, plants, bacteria and even other viruses. The 2019 novel coronavirus SARS-CoV-2 remains a constant threat to the human population. Viruses are biological objects with nanometric dimensions (typically from a few tens to several hundreds of nanometers). They are considered as the biomolecular substances composed of genetic materials (RNA or DNA), protecting capsid proteins and sometimes also of envelopes. Objective. The goal of the present review is to help predict the response and even destructuration of viruses taking into account the influence of different environmental factors, such as, mechanical loads, thermal changes, electromagnetic field, chemical changes and receptor binding on the host membrane. These environmental factors have significant impact on the virus.
Materials and methods. The study of viruses and virus-like structures has been analyzed using models and methods of nonlinear mechanics. In this regard, quantum, molecular and continuum descriptions in virus mechanics have been considered. Application of single molecule manipulation techniques, such as, atomic force microcopy, optical tweezers and magnetic tweezers has been discussed for a determination of the mechanical properties of viruses. Particular attention has been given to continuum damage–healing mechanics of viruses, proteins and virus-like structures. Also, constitutive modeling of viruses at large strains is presented. Nonlinear elasticity, plastic deformation, creep behavior, environmentally induced swelling (or shrinkage) and piezoelectric response of viruses were taken into account. Integrating a constitutive framework into ABAQUS, ANSYS and in-house developed software has been discussed. Conclusion. Link between virus structure, environment, infectivity and virus mechanics may be useful to predict the response and destructuration of viruses taking into account the influence of different environmental factors. Computational analysis using such link may be helpful to give a clear understanding of how neutralizing antibodies and T cells interact with the 2019 novel coronavirus SARS-CoV-2.
... Regarding the simulation of the viral entry into a cell membrane via endocytosis (Fig. 5), such important aspects were involved as the contact mechanics of viruses onto a flexible membrane [30,[82][83][84][85], effects of virus size and ligand density on the membrane wrapping [86], as well as, influence of receptor concentration gradient on the ligand-receptor binding [87]. The change of receptor density over the host membrane was described [37,88,89] by the Fick's second law [90][91][92], however, without consideration of diffusion induced stresses [93][94][95][96][97][98]. Continuum modeling was used for adhesive contact between the virus and cell membrane, driven by adhesion [82][83][84] or driven by adhesion and, additionally, by external displacement (or force) [30,85], however, under the assumption of linear elasticity. ...
Viruses are a large group of pathogens that have been identified to infect animals, plants, bacteria and even other viruses. The 2019 novel coronavirus SARS-CoV-2 remains a constant threat to the human population. Viruses are biological objects with nanometric dimensions (typically from a few tens to several hundreds of nanometers). They are considered as the biomolecular substances composed of genetic materials (RNA or DNA), protecting capsid proteins and sometimes also of envelopes. The study of viruses and virus-like structures has been analyzed using models and methods of nonlinear mechanics. In this regard, quantum, molecular and continuum descriptions in virus mechanics have been considered. Application of single molecule manipulation techniques, such as, atomic force microcopy, optical tweezers and magnetic tweezers has been discussed for a determination of the mechanical properties of viruses. Particular attention has been given to continuum damage–healing mechanics of viruses, proteins and virus-like structures. Also, constitutive modeling of viruses at large strains is presented. Nonlinear elasticity, plastic deformation, creep behavior, environmentally induced swelling (or shrinkage) and piezoelectric response of viruses were taken into account. The environment is considered to be made up of those parts of the universal system with which the virus interacts. Integrating a constitutive framework into ABAQUS, ANSYS and in-house developed software has been discussed. Link between virus structure, environment, infectivity and virus mechanics may be useful to predict the response and destructuration of viruses taking into account the influence of different environmental factors. Computational analysis using such link may be helpful to give a clear understanding of how neutralizing antibodies and T cells interact with the 2019 novel coronavirus SARS-CoV-2.
... 11 It was also shown that sometimes multiple receptors are needed for a virus to enter the cell, which might lead to even longer and more complex diffusion processes. 13 The problem of association of small ligands with specific active sites on the surfaces has been investigated before using various theoretical tools. [14][15][16][17][18][19][20] Analysis of this class of problems was initiated by Berg. ...
Trapping by active sites on surfaces plays important roles in various chemical and biological processes, including catalysis, enzymatic reactions, and viral entry into host cells. However, the mechanisms of these processes remain not well understood, mostly because the existing theoretical descriptions are not fully accounting for the role of the surfaces. Here we present a theoretical investigation on the dynamics of surface-assisted trapping by specific active sites. In our model, a diffusing particle can occasionally reversibly bind to the surface and diffuse on it before reaching the final target site. An approximate theoretical framework is developed, and its predictions are tested by Brownian Dynamics computer simulations. It is found that the surface diffusion can be crucial in mediating the association to trapping binding active sites. Our theoretical predictions work reasonably well as long as the size of the active site is much smaller than the overall surface area. Potential applications of our approach are discussed.
... 11 It was also shown that sometimes multiple receptors are needed for a virus to enter the cell, which might lead to even longer and more complex diffusion processes. 13 The problem of association of small ligands with specific active sites on the surfaces has been investigated before using various theoretical tools. 14-17 A diffusion limited association, when the reaction of binding to the surface (both specific and non-specific) is instantaneous, has been considered by Berg. ...
Trapping by active sites on surfaces plays important roles in various chemical and biological processes, including catalysis, enzymatic reactions, and viral entry into host cells. However, the mechanisms of these processes remain not well understood, mostly because the existing theoretical descriptions are not fully accounting for the role of the surfaces. Here we present a theoretical investigation on the dynamics of surface-assisted trapping by specific active sites. In our model, a diffusing particle can be occasionally reversibly bound to the surface and diffuse on it before reaching the final target site. An approximate theoretical framework is developed, and its predictions are tested by Brownian Dynamics computer simulations. It is found that the surface diffusion can be crucial in mediating the association to active sites. Our theoretical predictions work reasonably well as long as the size of the active site is much smaller than the overall surface area. Potential applications of our method are discussed.
... (1)]. The subsequent formation of tethers occurs via diffusion of receptors to the virion-cell-membrane contact region as, for example, described in Ref. [31]. This process is accompanied by the membrane bending and then by the virion entry into the cell occurring usually via endocytosis or, in the case of enveloped viruses, via fusion of the virion envelope with the cell membrane (reviewed in Refs. ...
Viral infections occur at very different length and time scales and include various processes which can often be described by using the models developed and/or employed in colloid and interface science. Bering in mind the currently active COVID 19, I discuss herein the models aimed at viral transmission via respiratory droplets and the contact of virions with the epithelium. In a more general context, I outline the models focused on penetration of virions via the cellular membrane, initial stage of viral genome replication, and formation of viral capsids in cells. In addition, the models related to a new generation of drug delivery vehicles, e.g., lipid nanoparticles with size about 100-200 nm, are discussed as well. Despite the high current interest in all these processes, their understanding is still limited, and this area is open for new theoretical studies.
... The experimental progress has parallely led to multiple theoretical models. A mathematical framework, based on probabilities for binding rates, introducing a random and a sequential driving mechanism for the receptors is provided in [16]. Discrete stochastic models for specific receptor-ligand adhesion as well as non-equilibrium continuum models for the competition between different modes of junction remodeling under force are developed in [17], whereas a statistical thermodynamic model of viral budding is presented in [18]. ...
The present study focuses on the receptor driven endocytosis typical of viral entry into a cell. A locally increased density of receptors at the time of contact between the cell and the virus is necessary in this case. The virus is considered as a substrate with fixed receptors on its surface, whereas the receptors of the host cell are free to move over its membrane, allowing a local change in their concentration. In the contact zone the membrane inflects and forms an envelope around the virus. The created vesicle imports its cargo into the cell. This paper assumes the diffusion equation accompanied by boundary conditions requiring the conservation of binders to describe the process. Moreover, it introduces a condition defining the energy balance at the front of the adhesion zone. The latter yields the upper limit for the size of virus which can be engulfed by the cell membrane. The described moving boundary problem in terms of the binder density and the velocity of the adhesion front is well posed and numerically solved by using the finite difference method. The illustrative examples have been chosen to show the influence of the process parameters on the initiation and the duration of the process.
... The experimental progress has parallely led to multiple theoretical models. A mathematical framework, based on probabilities for binding rates, introducing a random and a sequential driving mechanism for the receptors is provided in [16]. Discrete stochastic models for specific receptor-ligand adhesion as well as non-equilibrium continuum models for the competition between different modes of junction remodeling under force are developed in [17], whereas a statistical thermodynamic model of viral budding is presented in [18]. ...
The present study focuses on the receptor driven endocytosis typical of viral entry into a cell. A locally increased density of receptors at the time of contact between the cell and the virus is necessary in this case. The virus is considered as a substrate with fixed receptors on its surface, whereas the receptors of the host cell are free to move over its membrane, allowing a local change in their concentration. In the contact zone the membrane inflects and forms an envelope around the virus. The created vesicle imports its cargo into the cell. This paper assumes the diffusion equation accompanied by boundary conditions requiring the conservation of binders to describe the process. Moreover, it introduces a condition defining the energy balance at the front of the adhesion zone. The latter yields the upper limit for the size of virus which can be engulfed by the cell membrane. The described moving boundary problem in terms of the binder density and the velocity of the adhesion front is well posed and numerically solved by using the finite difference method. The illustrative examples have been chosen to show the influence of the process parameters on the initiation and the duration of the process.
... Furthermore, CD4/Lck signalling, while not essential for HIV infection (macrophages are Lck-negative yet HIV-susceptible), may play a more significant role in facilitating efficient HIV entry in some cell types than previously appreciated. A few theoretical studies have used stochastic models that account for receptor diffusion and receptor-virus binding to predict the accumulation of HIV receptors at virus-binding sites on the surface of target cells [72][73][74]. Future comparison of model predictions and experimental data may provide further insights into the mechanisms of receptor cluster formation. ...
The first step of cellular entry for the human immunodeficiency virus type-1 (HIV-1) occurs through the binding of its envelope protein (Env) with the plasma membrane receptor CD4 and co-receptor CCR5 or CXCR4 on susceptible cells, primarily CD4+ T cells and macrophages. Although there is considerable knowledge of the molecular interactions between Env and host cell receptors that lead to successful fusion, the precise way in which HIV-1 receptors redistribute to sites of virus binding at the nanoscale remains unknown. Here, we quantitatively examine changes in the nanoscale organisation of CD4 on the surface of CD4+ T cells following HIV-1 binding. Using single-molecule super-resolution imaging, we show that CD4 molecules are distributed mostly as either individual molecules or small clusters of up to 4 molecules. Following virus binding, we observe a local 3-to-10-fold increase in cluster diameter and molecule number for virus-associated CD4 clusters. Moreover, a similar but smaller magnitude reorganisation of CD4 was also observed with recombinant gp120. For one of the first times, our results quantify the nanoscale CD4 reorganisation triggered by HIV-1 on host CD4+ T cells. Our quantitative approach provides a robust methodology for characterising the nanoscale organisation of plasma membrane receptors in general with the potential to link spatial organisation to function.
... One such model simulated HIV entry by considering static viruses in the vicinity of freely diffusing receptors (311). A single receptor species was considered (without distinction between receptors and co-receptors) with a given diffusion coefficient and receptor concentration. ...
... The entry probability was found to increase with receptor mobility. For highly mobile receptors, the entry probability increased with N, whereas this trend was inverted for less mobile receptors (311). A Brownian dynamics model accounting for receptor concentration and diffusion on the cell surface as well as for Env density on the virus surface showed initial engagement of CD4 by Env trimers taking place within fractions of a second, to tether the virus to the cell surface. ...
As part of their entry and infection strategy, viruses interact with specific receptor molecules expressed on the surface of target cells. The efficiency and kinetics of the virus-receptor interactions required for a virus to productively infect a cell is determined by the biophysical properties of the receptors, which are in turn influenced by the receptors' plasma membrane (PM) environments. Currently, little is known about the biophysical properties of these receptor molecules or their engagement during virus binding and entry. Here we review virus-receptor interactions focusing on the human immunodeficiency virus type 1 (HIV), the etiological agent of acquired immunodeficiency syndrome (AIDS), as a model system. HIV is one of the best characterised enveloped viruses, with the identity, roles and structure of the key molecules required for infection well established. We review current knowledge of receptor-mediated HIV entry, addressing the properties of the HIV cell-surface receptors, the techniques used to measure these properties, and the macromolecular interactions and events required for virus entry. We discuss some of the key biophysical principles underlying receptor-mediated virus entry and attempt to interpret the available data in the context of biophysical mechanisms. We also highlight crucial outstanding questions and consider how new tools might be applied to advance understanding of the biophysical properties of viral receptors and the dynamic events leading to virus entry.
