Imperfect DNA lesion repair in the semiconservative quasispecies model: Derivation of the Hamming class equations and solution of the single-fitness peak landscape

This paper develops a Hamming class formalism for the semiconservative quasispecies equations with imperfect lesion repair, first presented and analytically solved in Y. Brumer and E.I. Shakhnovich (q-bio.GN/0403018, 2004). Starting from the quasispecies dynamics over the space of genomes, we derive an equivalent dynamics over the space of ordered sequence pairs. From this set of equations, we are able to derive the infinite sequence length form of the dynamics for a class of fitness landscapes defined by a master genome. We use these equations to solve for a generalized single-fitness-peak landscape, where the master genome can sustain a maximum number of lesions and remain viable. We determine the mean equilibrium fitness and error threshold for this class of landscapes, and show that when lesion repair is imperfect, semiconservative replication displays characteristics from both conservative replication and semiconservative replication with perfect lesion repair. The work presented here provides a formulation of the model which greatly facilitates the analysis of a relatively broad class of fitness landscapes, and thus serves as a convenient springboard into biological applications of imperfect lesion repair.

Figure 1

(Color online) Comparison between conservative and semiconservative replication. $\mathbf{P}$ denotes the parent strands, while $\mathbf{D}$ denotes the daughter strands.

Figure 2

(Color online) Illustration of the various definitions for ${\sigma }_{\left(N\right)C}$, ${\sigma }_{\left(N\right)C}^{″}$, and ${\sigma }_{\left(N\right)C}^{″}$. We use the four bases adenine $\left(A\right)$, guanine $\left(G\right)$, thymine $\left(T\right)$, and cytosine $\left(G\right)$. The Watson-Crick pairs are $A\text{−}T$ and $G\text{−}C$ [35]. The notations $5^{\prime }$ and $3^{\prime }$ refer to the chemically distinct ends of the polynucleotide chains [35].

Figure 3

(Color online) Diagram illustrating the definitions $l_C$, $l_L$, $l_R$, and $l_B$.

Figure 4

Comparison of theory and simulation results for $l=0$.

Figure 5

Comparison of theory and simulation results for $l=1$.

Figure 6

Comparison of theory and simulation results for $l=\infty$.

Figure 7

(Color online)Illustration of the various definitions $l_{C,1}$, $l_{C,2}$, $l_{C,3}$, $l_1^{″}$, $l_2^{″}$, and $l_3^{″}$.