Self-Assembly and Evolution of Homomeric Protein Complexes

We introduce a simple “patchy particle” model to study the thermodynamics and dynamics of self-assembly of homomeric protein complexes. Our calculations allow us to rationalize recent results for dihedral complexes. Namely, why evolution of such complexes naturally takes the system into a region of interaction space where (i) the evolutionarily newer interactions are weaker, (ii) subcomplexes involving the stronger interactions are observed to be thermodynamically stable on destabilization of the protein-protein interactions, and (iii) the self-assembly dynamics are hierarchical with these same subcomplexes acting as kinetic intermediates.

Figure 1

Schematic depiction of (a) a $D_2$ tetramer and the possible equilibria involved in its formation and (b) a $C_4$ tetramer.

Figure 2

(a) Free-energy diagram showing the dependence of the most stable state of the $D_2$ tetramer system on temperature and relative interaction strengths. The lines indicate where the equilibrium constants for the association reactions in Fig. 1a are one; lines are solid when the equilibrium is between the two most stable forms and dashed otherwise. The diagram is shaded according to whether monomers, dimers, or tetramers are most stable. The arrow indicates a possible evolutionary path from a dimeric to a tetrameric complex. (b) Dependence of the final yields of monomers, dimers, and tetramers from our dynamics simulations on $kT/{ϵ}_{AA}$ and ${ϵ}_{BB}/{ϵ}_{AA}$. Each pixel represents the result of a separate simulation, where each simulation started from a random configuration of the 400 particles and was ${10}^8$ steps long. (c) Equilibrium probability of a particle being monomeric or in a dimer or tetramer at ${ϵ}_{BB}/{ϵ}_{AA}=0.2$.

Figure 3

(a),(b) Time dependence of the probability of a particle being in a particular state for (a) ${ϵ}_{BB}/{ϵ}_{AA}=0.6$ and $kT/{ϵ}_{AA}=0.05$ and (b) ${ϵ}_{BB}/{ϵ}_{AA}=1$ and $kT/{ϵ}_{AA}=0.04$. (c) Dependence of $h$, the hierarchicity index, on temperature and relative interaction strengths. The state points corresponding to (a) and (b) are each marked by a white dot.

Figure 4

(a) Possible equilibria involved in the formation of a $D_4$ octamer. (b) Free-energy diagram showing the dependence of the most stable state on temperature and interaction strengths, superimposed on the final yields of monomers, dimers, tetramers, and octamers in our dynamics simulations (800 particles, ${10}^8$ steps). Note the change in the ordinate and abscissa in each half of the diagram.