Iterated function systems for DNA replication

The kinetic equations of DNA replication are shown to be exactly solved in terms of iterated function systems, running along the template sequence and giving the statistical properties of the copy sequences, as well as the kinetic and thermodynamic properties of the replication process. With this method, different effects due to sequence heterogeneity can be studied, in particular, a transition between linear and sublinear growths in time of the copies, and a transition between continuous and fractal distributions of the local velocities of the DNA polymerase along the template. The method is applied to the human mitochondrial DNA polymerase $\gamma$ without and with exonuclease proofreading.

Figure 1

${\mathrm{Exo}}^{-}$ human mitochondrial DNA polymerase: bulk probabilities and waiting times of the polymerase moving at the physiological concentrations (41, 42, 43, 44) and $\left[{\mathrm{PP}}_{\mathrm{i}}\right]={10}^{-4}$ M, along the Bernoulli template shown above. The theoretical results are depicted as crosses and those of Monte Carlo simulations with a statistics over ${10}^8$ copies as open symbols. Here, the mean growth velocity and the error probability take the values $v\simeq 18.6{\mathrm{s}}^{-1}$ and $\eta \simeq 1.13\times {10}^{-4}$, computed with the IFS method over a template of length $L={10}^5$.

Figure 2

${\mathrm{Exo}}^{-}$ human mitochondrial DNA polymerase: bulk probabilities and waiting times of the polymerase moving at the physiological concentrations (41), (42), (44), and $\left[{\mathrm{PP}}_{\mathrm{i}}\right]={10}^{-4}$ M, but $\left[{\mathrm{dGTP}}_{\mathrm{i}}\right]={10}^{-9}$ M, along the Bernoulli template shown above. The theoretical results are depicted as crosses and those of Monte Carlo simulations with a statistics over ${10}^8$ copies as open symbols. Here, the mean growth velocity and the error probability take the values $v\simeq 9.02\times {10}^{-2}{\mathrm{s}}^{-1}$ and $\eta \simeq 2.98\times {10}^{-2}$, computed with the IFS method over a template of length $L={10}^5$.

Figure 3

${\mathrm{Exo}}^{-}$ human mitochondrial DNA polymerase: bulk probabilities and waiting times of the polymerase moving at the physiological concentrations (41), (42), (44), and $\left[{\mathrm{PP}}_{\mathrm{i}}\right]={10}^{-4}$ M, but $\left[{\mathrm{dGTP}}_{\mathrm{i}}\right]=0$, along the Bernoulli template shown above. The theoretical results are depicted as crosses and those of Monte Carlo simulations with a statistics over ${10}^8$ copies as open symbols. Here, the mean growth velocity and the error probability take the values $v\simeq 7.50\times {10}^{-3}{\mathrm{s}}^{-1}$ and $\eta \simeq 0.355$, computed with the IFS method over a template of length $L={10}^5$.

Figure 4

${\mathrm{Exo}}^{-}$ human mitochondrial DNA polymerase: bulk probabilities and waiting times of the polymerase moving at the physiological concentrations (41), (42), (44), and $\left[{\mathrm{PP}}_{\mathrm{i}}\right]={10}^{-4}$ M, but $\left[\mathrm{dGTP}\right]={10}^{-3}$ M, along the Bernoulli template shown above. The theoretical results are depicted as crosses and those of Monte Carlo simulations with a statistics over ${10}^8$ copies as open symbols. Here, the mean growth velocity and the error probability take the values $v\simeq 1.54{\mathrm{s}}^{-1}$ and $\eta \simeq 7.88\times {10}^{-2}$, computed with the IFS method over a template of length $L={10}^5$.

Figure 5

${\mathrm{Exo}}^{-}$ human mitochondrial DNA polymerase at the pyrophosphate concentration $\left[{\mathrm{PP}}_{\mathrm{i}}\right]={10}^{-4}$ M: velocity $v$, affinity $A=\epsilon +D$, free-energy driving force $\epsilon$, conditional Shannon disorder $D$, error probability $\eta$, and entropy production rate $\mathrm{\Sigma }=Av$ versus equal nucleotide concentrations. The dots are obtained by Monte Carlo simulations generating ${10}^5$ copies of maximum length ${10}^5$. The lines depict the results of the IFS run over sequences of length ${10}^5$.

Figure 6

${\mathrm{Exo}}^{-}$ human mitochondrial DNA polymerase at the physiological concentrations (41), (42), (44) and pyrophosphate concentration $\left[{\mathrm{PP}}_{\mathrm{i}}\right]={10}^{-4}$ M: velocity $v$, affinity $A=\epsilon +D$, free-energy driving force $\epsilon$, conditional Shannon disorder $D$, error probability $\eta$, and entropy production rate $\mathrm{\Sigma }=Av$ versus the concentration $\left[\mathrm{dGTP}\right]$. The dots are obtained by Monte Carlo simulations generating ${10}^5$ copies of maximum length ${10}^5$. The lines depict the results of the IFS run over sequences of length ${10}^4$.

Figure 7

${\mathrm{Exo}}^{-}$ human mitochondrial DNA polymerase at the physiological concentrations (41, 42, 43, 44) and $\left[{\mathrm{PP}}_{\mathrm{i}}\right]={10}^{-4}$ M: probabilities of the four bases composing DNA strands of length ${10}^6$ versus the number of replications during the Monte Carlo simulation of successive replications. The evolution starts from a Bernoulli sequence with the equal probabilities of the four bases. The mean growth velocity and error probability are, respectively, $v\simeq 18.8$ and $\eta \simeq 1.1\times {10}^{-4}$.

Figure 8

${\mathrm{Exo}}^{-}$ human mitochondrial DNA polymerase at the concentrations $\left[\mathrm{dGTP}\right]={10}^{-9}$ M and $\left[{\mathrm{PP}}_{\mathrm{i}}\right]={10}^{-2}$ M: velocity $v$, affinity $A=\epsilon +D$, free-energy driving force $\epsilon$, conditional Shannon disorder $D$, error probability $\eta$, and entropy production rate $\mathrm{\Sigma }=Av$ versus the equal nucleotide concentrations $\left[\mathrm{dATP}\right]=\left[\mathrm{dCTP}\right]=\left[\mathrm{dTTP}\right]$. The dots are obtained by Monte Carlo simulations generating ${10}^5$ copies of maximum length ${10}^5$. The lines depict the results of the IFS run over sequences of length ${10}^5$.

Figure 9

${\mathrm{Exo}}^{-}$ human mitochondrial DNA polymerase at the pyrophosphate concentration $\left[{\mathrm{PP}}_{\mathrm{i}}\right]={10}^{-2}$ M, moving along a binary Bernoulli template composed of the two bases A and T: velocity $v$, affinity $A=\epsilon +D$, free-energy driving force $\epsilon$, conditional Shannon disorder $D$, error probability $\eta$, and entropy production rate $\mathrm{\Sigma }=Av$ versus the equal nucleotide concentrations $\left[\mathrm{dATP}\right]=\left[\mathrm{dTTP}\right]$. The dots are obtained by Monte Carlo simulations generating ${10}^5$ copies of maximum length ${10}^5$. The lines depict the results of the IFS run over sequences of length ${10}^5$.

Figure 10

${\mathrm{Exo}}^{-}$ human mitochondrial DNA polymerase at the pyrophosphate concentration $\left[{\mathrm{PP}}_{\mathrm{i}}\right]={10}^{-2}$ M, moving along a binary Bernoulli template composed of the two bases A and T: the distribution of the local partial velocity $v_{\mathrm{A}}$ versus the equal concentrations $\left[\mathrm{dATP}\right]=\left[\mathrm{dTTP}\right]$. The computation is carried out with the IFS method along a sequence of length ${10}^4$.

Figure 11

The distribution of the local velocity $x$ versus the equal concentrations $\left[\mathrm{dATP}\right]=\left[\mathrm{dTTP}\right]$ under the same conditions as in Fig. 10.

Figure 12

The fractal dimensions of the support of the distributions of the local partial velocities $\mathbf{v}$ ($\dim F_{\mathbf{v}}$) and the local velocity $x$ ($\dim F_x$) versus the equal concentrations $\left[\mathrm{dATP}\right]=\left[\mathrm{dTTP}\right]$ under the same conditions as in Fig. 10.

Figure 13

The IFS of the simple error-free model (72) at the concentrations $\left[\mathrm{dATP}\right]=\left[\mathrm{dTTP}\right]={10}^{-7}\mathrm{M}$, $\left[\mathrm{dCTP}\right]=\left[\mathrm{dGTP}\right]=0$, and $\left[{\mathrm{PP}}_{\mathrm{i}}\right]={10}^{-2}$ M. The iterations are shown giving the local velocities over a sequence containing 15 successive nucleotides. The fractal is shown in red along the horizontal and vertical axes.

Figure 14

${\mathrm{Exo}}^{+}$ human mitochondrial DNA polymerase at the concentrations $\left[{\mathrm{PP}}_{\mathrm{i}}\right]={10}^{-4}$ M and $\left[\mathrm{dNMP}\right]={10}^{-5}$ M: velocity $v$, affinity $A=\epsilon +D$, free-energy driving force $\epsilon$, conditional Shannon disorder $D$, error probability $\eta$, and entropy production rate $\mathrm{\Sigma }=Av$ versus the equal nucleotide concentrations $\left[\mathrm{dATP}\right]=\left[\mathrm{dCTP}\right]=\left[\mathrm{dGTP}\right]=\left[\mathrm{dTTP}\right]$. The dots are obtained by Monte Carlo simulations generating ${10}^5$ copies of maximum length ${10}^5$. The lines depict the results of the IFS run over sequences of length ${10}^5$.

Figure 15

${\mathrm{Exo}}^{+}$ human mitochondrial DNA polymerase at the concentrations $\left[\mathrm{dGTP}\right]={10}^{-9}$ M, $\left[{\mathrm{PP}}_{\mathrm{i}}\right]={10}^{-4}$ M, and $\left[\mathrm{dNMP}\right]={10}^{-5}$ M: velocity $v$, affinity $A=\epsilon +D$, free-energy driving force $\epsilon$, conditional Shannon disorder $D$, error probability $\eta$, and entropy production rate $\mathrm{\Sigma }=Av$ versus the equal nucleotide concentrations $\left[\mathrm{dATP}\right]=\left[\mathrm{dCTP}\right]=\left[\mathrm{dTTP}\right]$. The dots are obtained by Monte Carlo simulations generating ${10}^5$ copies of maximum length ${10}^5$. The lines depict the results of the IFS run over sequences of length ${10}^5$.