A Family of Fitness Landscapes Modeled through Gene Regulatory Networks
Abstract
:1. Introduction
2. Methods
2.1. Pathway Framework of GRNs
2.2. Fitness Landscape of GRNs under the Pathway Framework
3. Results
3.1. Connectivity and Accessibility in a Fitness Landscape of GRNs
3.2. Mesoscopic Skeleton Derived from “Symmetries” in the Genotype Network of GRNs
 (iii)
 Change the source node of an edge from one stimulus to another stimulus and vice versa, e.g., in Figure 4d, moving an edge pointing from node 1 to node 3 to pointing from node 2. (Note that this operation is not necessarily equivalent to permuting the identities of stimuli since at most only the single focal edge will be affected.)
 (iv)
 Move a selfloop at one node to another node and vice versa, for example, reallocating a selfloop at node 3 to node 4 in Figure 4e.
3.3. Algorithmic Construction of the Mesoscopic Backbone of GRN Fitness Landscape
 If ${g}^{\prime}$ has one more nonselfloop edge than g, then ${g}^{\prime}\in {\mathcal{M}}^{+}\left(g\right)$;
 If ${g}^{\prime}$ has one less nonselfloop edge than g, then we have $g\in {\mathcal{M}}^{+}\left({g}^{\prime}\right)$;
 If ${g}^{\prime}$ has the same number of nonselfloop edges as g, and then they share a common mutational neighbor ${g}^{\u2033}$, where the only different edge between g and ${g}^{\prime}$ is rewired to a selfloop and thus $g,{g}^{\prime}\in {\mathcal{M}}^{+}\left({g}^{\u2033}\right)$.
 (A)
 For an equivalence class $\theta \in {\mathsf{\Theta}}_{k}$ and its representative GRN $g\in \theta $, under what condition will ${g}_{1}^{\prime},{g}_{2}^{\prime}\in {\mathcal{M}}^{+}\left(g\right)$ belong to the same equivalence class in layer ${\mathsf{\Theta}}_{k+1}$?
 (B)
 For two distinct equivalence classes ${\theta}_{1},{\theta}_{2}\in {\mathsf{\Theta}}_{k}$ and their representative GRNs ${g}_{1}\in {\theta}_{1}$ and ${g}_{2}\in {\theta}_{2}$, under what condition will ${g}_{1}^{\prime}\in {\mathcal{M}}^{+}\left({g}_{1}\right)$ and ${g}_{2}^{\prime}\in {\mathcal{M}}^{+}\left({g}_{2}\right)$ belong to the same equivalence class in layer ${\mathsf{\Theta}}_{k+1}$?
 There is an integer p such that ${\pi}^{p}\left({\gamma}_{1}\right)={\gamma}_{1}$ and $({\pi}^{p}\left({u}_{1}\right),{\pi}^{p}\left({v}_{1}\right))=({u}_{1},{v}_{1})$;
 There is another integer $q<p$ such that ${\pi}^{q}\left({\gamma}_{1}\right)={\gamma}_{2}$ and $({\pi}^{q}\left({u}_{1}\right),{\pi}^{q}\left({v}_{1}\right))=({u}_{2},{v}_{2})$;
 ${e}_{{g}_{2}^{\prime}}\left({\pi}^{k}\left({\gamma}_{1}\right)\right)=({\pi}^{k}\left({u}_{1}\right),{\pi}^{k}\left({v}_{1}\right))$ for $k=1,2,\dots ,q$;
 ${e}_{{g}_{2}^{\prime}}\left({\pi}^{k}\left({\gamma}_{1}\right)\right)\ne ({\pi}^{k}\left({u}_{1}\right),{\pi}^{k}\left({v}_{1}\right))$ for $k=q+1,q+2,\dots ,p$;
 For any locus $\gamma $ and nonselfloop source–target pair $(u,v)$ such that $(\gamma ,u,v)\ne ({\pi}^{k}\left(\gamma \right),{\pi}^{k}\left({u}_{1}\right),{\pi}^{k}\left({v}_{1}\right))$ for $0\le k\le q1$, we have ${e}_{g}\left(\pi \left(\gamma \right)\right)=(\pi \left(u\right),\pi \left(v\right))$ if and only if ${e}_{g}\left(\gamma \right)=(u,v)$.
 (I)
 For every representative GRN g in ${\mathsf{\Theta}}_{k}$ and every phenotypepreserving automorphism $\sigma $ of g, there is an operation ${\psi}_{g,\sigma}$ that joins together the groups of ${g}_{1}^{\prime}=g\oplus (\gamma ,{u}_{1},{v}_{1})$ and ${g}_{2}^{\prime}=g\oplus (\gamma ,{u}_{2},{v}_{2})$, where ${u}_{1},{u}_{2}\in {\mathsf{\Omega}}_{0}$ and ${v}_{2}=\sigma \left({v}_{1}\right)$;
 (II)
 For every representative GRN g in ${\mathsf{\Theta}}_{k}$ and every phenotypepreserving automorphism $\overline{\sigma}$ of each subgraph $\overline{g}$ of g such that the edge differences ${\mathsf{\Gamma}}^{\prime}\left(g\right)\u29f5{\mathsf{\Gamma}}^{\prime}\left(\overline{g}\right)$ are sequentially connected via $\overline{\sigma}$, there is an operation ${\varphi}_{g,\overline{g},\overline{\sigma}}$ that joins together the groups of ${g}_{1}^{\prime}=g\oplus ({\gamma}_{1},{u}_{1},{v}_{1})$ and ${g}_{2}^{\prime}=g\oplus ({\gamma}_{2},{u}_{2},{v}_{2})$, where automorphism $\overline{\sigma}$ consecutively transforms edge ${\gamma}_{1}$ into ${\gamma}_{2}$ through ${\mathsf{\Gamma}}^{\prime}\left(g\right)\u29f5{\mathsf{\Gamma}}^{\prime}\left(\overline{g}\right)$;
 (III)
 For every representative GRN ${g}^{\u2033}$ in ${\mathsf{\Theta}}_{k1}$ and each ${\tilde{g}}_{1}={g}^{\u2033}\oplus ({\gamma}_{1}^{\u2033},{u}_{1}^{\u2033},{v}_{1}^{\u2033})$ and ${\tilde{g}}_{2}={g}^{\u2033}\oplus ({\gamma}_{2}^{\u2033},{u}_{2}^{\u2033},{v}_{2}^{\u2033})$ in two different equivalence classes ${\theta}_{1}$ and ${\theta}_{2}$, such that we have phenotypepreserving isomorphisms ${\pi}_{1}$/${\pi}_{2}$ from ${\tilde{g}}_{1}$/${\tilde{g}}_{2}$ to the representative GRN ${g}_{1}$/${g}_{2}$ after selfloop removal, there is an operation ${\phi}_{g,{\tilde{g}}_{1},{\tilde{g}}_{2}}$ that joins together the groups of ${g}_{1}^{\prime}={g}_{1}\oplus ({\pi}_{2}\left({\gamma}_{2}^{\u2033}\right),{\pi}_{2}\left({u}_{2}^{\u2033}\right),{\pi}_{2}\left({v}_{2}^{\u2033}\right))$, and ${g}_{2}^{\prime}={g}_{2}\oplus ({\pi}_{1}\left({\gamma}_{1}^{\u2033}\right),{\pi}_{1}\left({u}_{1}^{\u2033}\right),{\pi}_{1}\left({v}_{1}^{\u2033}\right))$.
Algorithm 1 Constructing the underlying space of a fitness landscape of GRNs 
Require: The fixed underlying collections of loci $\mathsf{\Gamma}$ and proteins $\mathsf{\Omega}$ of GRNs Ensure: The representative GRN ${g}_{\theta}$ of each equivalence class $\theta \in \mathsf{\Theta}$, and its number of mutational neighbors ${A}_{{g}_{\theta}}\left({\theta}^{\prime}\right)$ in any equivalence class ${\theta}^{\prime}\in \mathsf{\Theta}$

4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A. Central and Peripheral GRNs Where No Regulation Presents
Appendix B. PhenotypePreserving Automorphisms of the Genotype Network of GRNs
Appendix C. Combining Mutational Neighbors into Equivalence Classes
 (A)
 For an equivalence class $\theta \in {\mathsf{\Theta}}_{k}$ and its representative GRN $g\in \theta $, under what condition will ${g}_{1}^{\prime},{g}_{2}^{\prime}\in {\mathcal{M}}^{+}\left(g\right)$ belong to the same equivalence class in layer ${\mathsf{\Theta}}_{k+1}$?
 (B)
 For two distinct equivalence classes ${\theta}_{1},{\theta}_{2}\in {\mathsf{\Theta}}_{k}$ and their representative GRNs ${g}_{1}\in {\theta}_{1}$ and ${g}_{2}\in {\theta}_{2}$, under what condition will ${g}_{1}^{\prime}\in {\mathcal{M}}^{+}\left({g}_{1}\right)$ and ${g}_{2}^{\prime}\in {\mathcal{M}}^{+}\left({g}_{2}\right)$ belong to the same equivalence class in layer ${\mathsf{\Theta}}_{k+1}$?
 i.
 $({\sigma}^{q}\left({\gamma}_{1}\right),{\sigma}^{q}\left({u}_{1}\right),{\sigma}^{q})=({\gamma}_{2},{u}_{2},{v}_{2})$;
 ii.
 $({\sigma}^{p}\left({\gamma}_{1}\right),{\sigma}^{p}\left({u}_{1}\right),{\sigma}^{p}\left({v}_{1}\right))=({\gamma}_{1},{u}_{1},{v}_{1})$;
 iii.
 ${\left(\right)}_{{\sigma}^{k}}^{\left({\gamma}_{1}\right)}k=1q1$ and ${e}_{g}\left(\right)open="("\; close=")">{\sigma}^{k}\left({\gamma}_{1}\right)$ for $k=1,2,\dots ,q1$;
 iv.
 ${e}_{g}\left(\right)open="("\; close=")">{\sigma}^{k}\left({\gamma}_{1}\right)$ for $k=q,q+1,\dots ,p$.
 i.
 ${\tilde{g}}_{1}$ and ${g}_{1}$ belong to the same equivalence class;
 ii.
 ${\tilde{g}}_{1},{g}_{2}\in {\mathcal{M}}^{+}\left({g}^{\u2033}\right)$;
 iii.
 ${g}_{2}^{\prime}\in {\mathcal{M}}^{+}\left({\tilde{g}}_{1}\right)$.
Appendix D. Size of an Equivalence Class of GRNs
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Yang, C.H.; Scarpino, S.V. A Family of Fitness Landscapes Modeled through Gene Regulatory Networks. Entropy 2022, 24, 622. https://doi.org/10.3390/e24050622
Yang CH, Scarpino SV. A Family of Fitness Landscapes Modeled through Gene Regulatory Networks. Entropy. 2022; 24(5):622. https://doi.org/10.3390/e24050622
Chicago/Turabian StyleYang, ChiaHung, and Samuel V. Scarpino. 2022. "A Family of Fitness Landscapes Modeled through Gene Regulatory Networks" Entropy 24, no. 5: 622. https://doi.org/10.3390/e24050622