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From 2012.igem.org

Modelling

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 has a certain number of subunits or monomers. 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. But for the sake of modelling, we take an average of it forming 36 pentameric capsomeres.

For modelling our VLPs we needed to find a model which could fit the assembly of our entire virus like particles, namely, CCMV, HBV and PLRV. 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, where the self-association reaction begins with a nucleation event followed by elongation through sequential addition of free subunits, one at a time, to nuclei and partially assembled intermediates, until a closed icosahedral VLP is formed. Nucleus formation is regarded as the rate-limiting step.

The effective concentration of capsomeres participating in self-association, C, is defined as:

(1) C = C_total – C_critical

C_total is the total capsomere concentration and C_critical is the critical capsomere concentration. The critical concentration is defined as the concentration below which VLP formation will not take place.

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.

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

where C₂ is the dimer (nucleus) concentration, k_V,crit (M^-1 s^-1) the nucleation rate constant and k_V (M^-1 s^-1) the elongation rate constant.

The aggregation of the viral protein is modelled by a second-order reaction describing the agglomeration of two capsomeres:

(3) dA/dt = (1/2) * k_A * C₂

where A is the aggregate concentration and k_A (M^-1 s^-1) the aggregation rate constant.

The competing reactions of VLP assembly and aggregation can thus be described by the following set of equations:

(4) dC/dt = k_V,critical * C₂ - k_V * C * ∑_(i=2)^(s-1) C_i - k_A * C₂

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

(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.

Parameters to be determined were C_critical, K_v, K_v,critical and k_A