A kinetic Zipper model and the assembly of tobacco mosaic virus

Abstract

We put forward a modified Zipper model inspired by the statics and dynamics of the spontaneous reconstitution of rodlike tobacco mosaic virus particles in solutions containing the coat protein and the single-stranded RNA of the virus. An important ingredient of our model is an allosteric switch associated with the binding of the first protein unit to the origin-of-assembly domain of the viral RNA. The subsequent addition and conformational switching of coat proteins to the growing capsid we believe is catalyzed by the presence of the helical arrangement of bound proteins to the RNA. The model explains why the formation of complete viruses is favored over incomplete ones, even though the process is quasi-one-dimensional in character. We numerically solve the relevant kinetic equations and show that time evolution is different for the assembly and disassembly of the virus, the former exhibiting a time lag even if all forward rate constants are equal. We find the late-stage assembly kinetics in the presence of excess protein to be governed by a single-exponential relaxation, which agrees with available experimental data on TMV reconstruction.

Figure 1

Schematic representation of our kinetic zipper model for the assembly of Tobacco Mosaic Virus from coat proteins onto RNA templates with q binding sites. Nucleation of the assembly of the rodlike virus particles starts at the origin of assembly (OAS) region and is speculated to occur by either of two mechanisms: (A) by conformational switching of the 20S disk aggregate to a helix, which requires an energy h, to the OAS. (B) Alternatively, nucleation takes place by binding of the 20S helical aggregate with the RNA and ordering of the inner loops of the proteins making up the aggregate, which requires an energy h. Subsequent elongation happens by binding of (C) larger units such as the disk or the helical 20S protein aggregate or (D) the much smaller A-protein. Binding of the protein units to the RNA yields a binding free energy g, and the binding between two protein disks is associated with an interaction free energy ϵ. This guarantees sequential assembly commencing at the OAS.

Figure 2

Ratio of probabilities P(n)/P(0) for an RNA molecule to be coated by n protein units as a function of the affinity s (A) in the absence of allostery (σ = 1), and (B) in the presence of allosteric effects (σ = 10⁻³). The average fraction of binding sites occupied by protein aggregates, 〈θ〉, for (C) σ = 1 and (D) σ = 10⁻³ also as a function of the affinity s. For the number of binding sites on each polymeric template we set q = 63. (E and F) The average level of coverage of RNA strands by coat proteins 〈θ〉 as a function of the bare affinity s (E) in the absence of allostery (σ = 1), and (F) in the presence of allosteric effects (σ = 10⁻³). The effective number of binding sites equals q = 63.

Figure 3

(A) Assembly and (B) disassembly kinetics of TMV as modeled by a kinetic zipper for different values of the allosteric parameter σ presuming 63 binding sites for each RNA molecule. The quenches are between average surface coverages 〈θ〉 = 0.001 → 〈θ〉 = 0.9 and 〈θ〉 = 0.9 → 〈θ〉 = 0.001. (A, Inset) Expanded view of the onset of assembly, to point out the lag time introduced by the allosteric factor σ. (C) Assembly after a sudden quench from affinities s₁ = 0.35 → s₂ = 2.6 and (D) disassembly after the quench s₂ = 2.6 → s₁ = 0.35. Shown are the probabilities P(n) of n protein units on RNA molecules consisting of 63 binding sites as a function of the dimensionless time τ. Here, s₁ = 0.35 and s₂ = 2.6 correspond to a coverage of 〈θ〉 = 10⁻⁴ and 〈θ〉 = 99, respectively, and the allosteric parameter σ = 0.01. All on-rates are equal, implying κ = 1.

Figure 4

Comparison of the predictions of the kinetic zipper model with experimental data. (A) Experiments of Butler and Finch (44) on the reconstitution kinetics of TMV. Shown are fractions in percentile units of binned length ranges as a function of dimensionless time τ. To obtain a good fit, we set the model parameters described in the main text at σ = 0.001 and κ = 1, and presumed a quench from s₁ = 0.84 to s₂ = 2.58. (B and C) Comparison between experimental results probing some average of the size of the growing virus particles in TMV assembly experiments and the kinetic zipper model. (B) Radius of gyration R,_g,z, as obtained by SAXS versus time. (Symbols) Experimental data (35). (Drawn line) Theory (k₊ (σ = 1) = 40[min⁻¹], k₊ (σ = 0.001) = 65[min⁻¹]). (Inset) Enlarged part of the same plot. (C) Assembly kinetics followed by the increase in turbidity upon mixing TMV protein disks with TMV RNA. (Symbols) Experimental data (45). (Drawn line) Theory (k₊ (σ = 1) = 60[min⁻¹], k₊ (σ = 0.001) = 112[min⁻¹]). Model parameters for panels B and C were set to σ = 0.001, κ = 1. A quench from 〈θ〉 = 0 to 〈θ〉 = 0.9 was presumed.