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Contents 1 Mathemathic VLPs assembly model 1.1 Introduction 1.2 VLP Formation 2 References

Mathemathic VLPs assembly model

Introduction

In our project there are multiple aspects that we can model. Starting at the modelling of the formation of VLPs under different conditions, continuing at a dispersion model for when VLPs are injected into a human body and ending at marcro-models for degredation of VLPs in the environment. We chose to model VLP formation and aggregate formation.

VLP Formation

A VLP, which is a virus capsid without any genetic material, consists of a certain number of subunits or monomers. These subunits are viral proteins, which for our project are made to be expressed by E.coli and then made to re-assemble in vitro.

For modelling our VLPs we needed to find a model which could fit the assembly of our entire array of virus like particles, namely, CCMV, HBV and Polerovirus.In the case of CCMV, there are 180 subunits which go on to form capsomeres in the form of pentamers or hexamers and self-assemble according to the triangulation number, T=7 to form a capsid approximately 28nm in diameter. The CCMV capsid consists of 12 pentameric capsomeres and 20 hexagonal ones [2]. But for the sake of modelling, we take an average of it forming 36 pentameric capsomeres.

Presently, assembly of virus particles is not fully understood and different idealized models and theories are present which try to explain the assembly process. One such theory is the classical nucleation theory. In this theory, the first step is the self-association reaction which begins with a nucleation event (the formation of a dimer). This dimer then serves as the basis for elongation through sequential addition of free subunits, one at a time, to nuclei (dimer) to form partially assembled intermediates which results in a closed icosahedral VLP as shown in figure 7. Nucleus formation is regarded as the rate-limiting step [3].

We chose this model because it competitively accounts for the partitioning of the protein subunits between correct and the aggregation pathways. Using this, we can estimate losses through aggregate formation and also estimate the yield of the VLPs once the parameters have been determined.

The critical concentration of capsomeres is the concentration below which no VLP formation will take place. Therefore, to be able to form VLP's, the total concentration of capsomeres should be higher than the critical concentration. To establish this, we assume the concentration of capsomeres available for VLP formation (C) to be equal to the total concentration (C_total) minus the critical concentration (C_critical)[3]:

(1) C = C_total – C_critical

It is considered that a dimer is the first species (nucleus) formed in the reaction, and is subsequently consumed to form higher-order intermediates by the addition of free subunits (capsomeres). Two capsomeres will go on to form a dimer (rate limiting step), whose concentration is defined by C₂. The rate of change of concentration of dimers with time, depends on k_v,critical (M^-1 s^-1) which is the nucleation rate constant and k_v (M^-1 s^-1) the elongation rate constant, given by the equation [3]:

(2) dC₂/d_t = (1/2) * k_v,critical * C^2 – k_v * C * C₂

The aggregation of the viral protein is modelled by a second-order reaction describing the agglomeration of two capsomeres, where A is the aggregate concentration and k_A (M-1 s-1) the aggregation rate constant[3].

(3) dA/dt = (1/2) * k_A * C^2

The capsomere concentration will continuously drop, depending on the rate of dimer formation, elongation and aggregate formation. This equation is given by [3]:

(4) dC/dt = k_V,critical * C^2 - k_V * C * ∑_(i=2)^(s-1)[C_i] - k_A * C^2

On the path to the formation of the VLP, due to elongation intermediates are formed, with the first one (3 capsomeres long) forming after the elongation of a dimer. The equation for the formation of intermediates, with i being the number of subunits in a given intermediate, is given by [3]:

(5) dC_i/dt = k_V *C * (C_i-1 - C_i); i= 3; 4; . . . ; s-1

The last step of the process is the formation of the VLP. The VLP concentration is the end result of the elongation to the biggest possible intermediate (s-1), s being the number of subunits in a correctly formed VLP[3]:

(6) dV/dt = k_V * C * C_s-1

where V is the VLP concentration, s is the number of subunits in a correctly formed VLP (36 in the case of CCMV), and i is the number of subunits in a given intermediate. The initial conditions at t=0 were set as C=C₀ and C₂ = C₃ =...... = C₃₅= V=A=0.

The initial conditions at t=0 are C=C0 and C2 = C3 =...... = C35= V=A=0.

For our model, we ran simulations using MATLAB 2012. Values for the parameters, C_critical = 0.02 mg/ml, k_V,critical =2 ,k_V = 1000 and k_A = 10 were assumed. It should be noted that, since the rate constants would be in M^-1 s^-1 and the concentration in mg/ml, an intermediate C_i would weigh i times the weight of a capsomere. So, in the end, the VLP would weigh s times the weight of a capsomere. This was introduced into our model script and the resulting simulation (Figure 8) was obtained.

Our model predicts that for the assumed set of parameters, almost 75% of the capsomeres form VLPs and the rest go on to form aggregates after 8 hours.

Since the model depends on the total number of subunits in a VLP and certain parameters, it can be applied to the self-assembly of any virus-like particle composed of one kind of viral protein. The parameters can then be determined by incorporating experimental data measuring the concentration profiles of the proteins involved under differing conditions and analyzing the components put through the required purification steps. All this information can be combined with the model in MATLAB to give true values for the parameters. Due to a combination of time constraints and unavailability of instruments like asymmetrical flow-field flow fractionator, bioanalyzer etc to purify and measure concentrations of capsomere, VLPs and aggregates effectively we could not achieve this.

Self-assembly of VLPs occurs at certain conditions (pH, salt concentration, temperature). This self-assembly then is dependent on the rate constants (k_V,critical,k_V and k_A) involved in the dynamics. In theory, the elongation rate constant k_V should be much larger than k_V,critical and k_A. This would drive the system towards formation of VLPs. Also, ideally the k_A should be smaller than the k_V,critical which would mean, dimer formation is more dominant than aggregation, but this is rarely the case. So, ultimately, if one has the required instruments at their beck and call, the parameters could easily be determined and VLP production could be maximized and losses through aggregation, minimized.

The script for the MATLAB can be found here File:Assembly script.txt

References

1. Beterams, G., Böttcher B., & Nassal M. (15. Sept 2000). Packaging of up to 240 subunits of a 17 kDa nuclease into the interior of recombinant hepatitis B virus capsids. FEBS Lett., S. 169-76.

2.Johnson, J. E. and J. A. Speir (1997). "Quasi-equivalent viruses: a paradigm for protein assemblies." J Mol Biol 269(5): 665-675.

3.Ding, Y., Y. P. Chuan, et al. (2010). "Modeling the competition between aggregation and self-assembly during virus-like particle processing." Biotechnol Bioeng 107(3): 550-560.