Effects of local mutations in quadratic iterations

Abstract

We introduce mutations in replication systems in which the intact copying mechanism is performed by discrete iterations of a complex quadratic map in the family $f_c(z) = z^2+c$. More specifically, we consider a "correct" function $f_{c_1}$ acting on the complex plane (representing the RNA to be copied). A "mutation" $f_{c_0}$ is a different ("erroneous") map acting on a locus of given radius $r$ around a mutation focal point $ξ^*$. The effect of the mutation is interpolated radially to eventually recover the original map $f_{c_1}$ when reaching an outer radius $R$. We call the resulting map a "mutated" map. In the theoretical framework of mutated iterations, we study how a mutation (replication error) affects the temporal evolution of the system, in the context of cellular differentiation. We use the prisoner set of the system to quantify simultaneously the long-term behavior of the entire space under mutated maps. We analyze how the position, timing and size of the mutation can alter the system's long-term evolution (as encoded in the topology of its prisoner set). In the context of genetics, this framework may increase our understanding of the factors and mechanisms that shape the genetic expression, in a specialized cell, in the process of differentiation from a stem cell.

Figures (14)

• Figure 1: Prisoner sets corresponding to intact mutations for the generating parameters $c_1=0$ (unit disc), $c_1=-0.65$ (basilica) and $c_1=-0.13+0.77i$ (Douady rabbit). These can be interpreted as being the gene expression loci for three different types of cells, with different morphology and function (e.g., epithelial cell, kidney cell and neuron). In each panel, the prisoner set is shown in black (the points that never escape). The other colors represent the number of steps in which each point in $\mathbb{C}$ escapes the disc or radius $R_e=2$, under iterations of the corresponding map $f_{c_1}$ (in our interpretation, this represents the replication step where the gene stops being carried forward in the expression locus). The same color map will be used in all our other illustrations of prisoner sets, which are all computed in $800 \times 800$ resolution, and based on 100 total iterations.
• Figure 2: Mutation applied at the origin, illustrating the mutation focus, mutation disk and transition annulus, and showing the corresponding formulas for the iterated function $z \to f(z)$ on each portion of the complex plane.
• Figure 3: Example of prisoner sets for different mutations. The top panel illustrates the prisoner set of the intact map $f_{c_1}$, for $c_1=-0.13+0.77i$ (the Douady rabbit). The next row represents the prisoner sets for four different functions $f_{c_0}$, from left to right: $c_0=0$ (unit disk); $c_0=-0.65$ (basilica); $c_0=-0.117-0.856i$; $c_0=-i$ (dendrite). The two additional panels in each column illustrate the prisoner set for a system with a point-wise mutation $c_0$ at the origin, and transition radius $R=0.1$ (third row) and $R=0.5$ (fourth row).
• Figure 4: Dependence of prisoner set on the mutation radius $r$ when $D(R) \subseteq {\cal P}(f_{c_1})$. All panels represent mutations with fixed transition radius $R=0.1$, acting at the origin on $c_1= -0.13+0.77i$. From top to bottom, each row corresponds to a different mutant $c_0$, as follows: Top:$c_0=0$. Middle:$c_0=-0.13-0.77i$. Botom:$c_0=0.33$. The mutation radius is increased from left to right, so that each column corresponds respectively to: $r=0$; $r=0.04$; $r=0.08$; $r=0.1=R$.
• Figure 5: Examples of limit behavior of ${\cal P}(f)$ as $r \nearrow R$.Left.$c_1= -0.13+0.77i$, $c_0=0$. The transition radius $R$ is represented by the black circle, and is taken such that the disk $D(R)$ is a subset of the Douady rabbit ${\cal P}(f_{c_1})$ (shown as a contour). The orbit of an arbitrary point $z_0$ in $\overset{\circ}{\cal P}(f_{c_1})$, is sketched symbolically up to $z_N = f^{\circ N}_{c_1}(z_0)$, its first iterate that falls in the interior of $D(R)$. Then the mutation radius (represented by the red circle) can be chosen so that $\lvert z_N \rvert < r <R$, and $z_N \in D(r)$. The rest of the trajectory $z_{n}=f^{\circ n}(z_0)$, $n \geq N+1$ is then iterated under $f_{c_0}(z) = z^2$, and is therefore trapped in $D(r)$. Right.$c_1= -0.13+0.77i$, $c_0=-0.13-0.77i$. The contour of the Douady rabbit $c_1$ is shown in black, and that of the inverted rabbit $c_0$ is shown in gray. The transition radius $R$ is represented by the black circle, and is small enough so that $D(R) \subset {\cal P}(f_{c_1})$, and so that the first iterate of $D(R)$ (contained inside the green circle) is outside of ${\cal P}(f_{c_1})$, such that $f_{c_1}(D(R)) \cap {\cal P}(f_{c_0}) = \phi$. The orbit of an arbitrary $z_0 \in \overset{\circ}{\cal P}(f_{c_1})$ is shown symbolically up to its first iterate $z_N \in D(R)$. Then the mutation radius $r$ (red circle) can be chosen such that $\lvert z_N \rvert < r <R$, and $z_{N+1} = f(z_N)$ is outside of both $D(R)$ and ${\cal P}(f_{c_1})$, hence it escapes under $f$.

Theorems & Definitions (14)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Theorem 2.5
• Lemma 3.1
• Conjecture 3.2
• Theorem 3.3