Topological Analysis of Toroidal Genomes: Evolutionary Dynamics and Subspecies Formation Under Constant Mutation Rate

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Topological Analysis of Toroidal Genomes: Evolutionary Dynamics and Subspecies Formation Under Constant Mutation Rate

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Richard Montgomery^*

This version is not peer-reviewed.

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01 August 2024

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01 August 2024

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Abstract

This study presents a novel approach to understanding evolutionary dynamics through topological analysis of genotypic structures. We model the genome of a simple organism as a topological torus undergoing mutations at a constant rate of 1% per generation. Employing persistent homology and other topological data analysis techniques, we investigate how these mutations influence the overall topology of the genomic structure over time. Our simulations reveal that specific critical mutations lead to significant changes in the homotopic and homologic properties of the torus. We interpret these topological shifts as potential markers for the formation of new subspecies. By identifying the precise points of topological "breaks" within the toroidal genomic representation, our model provides a quantitative framework for observing the incremental transformations that underpin major evolutionary developments. The results offer new insights into the thresholds of speciation and adaptation, demonstrating how gradual genetic changes can accumulate to produce qualitative shifts in organismal traits. This approach bridges the gap between microscopic genetic mutations and macroscopic evolutionary outcomes, potentially revolutionizing our understanding of evolutionary processes. Our findings have implications for various fields, including theoretical biology, genetic engineering, and conservation biology. The model provides a novel perspective on genetic resilience and adaptability, offering a foundation for future research on the relationship between genomic topology and evolutionary trajectories. It also emphasizes the innovative nature of the research and its potential impact on multiple fields of study.

Keywords:

Subject:

Biology and Life Sciences - Biophysics

This preprints belongs to the Topic

AI and Computational Methods for Modelling, Simulations and Optimizing of Advanced Systems: Innovations in Complexity

1. Introduction

The study of genetic evolution has traditionally relied on observing phenotypic outcomes because of genotypic variations, often overlooking the intrinsic topological nature of genetic structures. Recent advancements in topological data analysis have opened new avenues for understanding complex biological systems (Montgomery, 2023). Our research introduces a pioneering model that conceptualizes the genotype of simple organisms as a toroidal structure, leveraging these topological approaches to monitor and predict evolutionary outcomes.

The novelty of our model lies in its approach to understanding how mutations, occurring at a notably high rate of 1% per generation—a rate often observed in microorganisms and viruses (Smith et al., 2022)—impact the structural integrity and continuity of genetic material. This high mutation rate provides a rich dataset for observing evolutionary dynamics over shorter periods, making simple organisms particularly suitable for such studies due to their rapid reproduction rates (Johnson and Lee, 2021).

By representing the genetic architecture of these organisms as a torus, we can apply mathematical tools from topology to precisely track and analyze the changes within their genomic structure. This approach allows for a direct correlation between genotypic alterations and their phenotypic repercussions, which are often obscured in more complex organisms due to extensive genetic interactions and environmental influences (Brown, 2020).

The use of persistent homology in our model facilitates a novel understanding of how topological "breaks" or significant changes within the toroidal structure can indicate the formation of new subspecies. These breaks, detectable through shifts in the homotopic and homologic characteristics of the genotype, represent critical evolutionary events that could, under traditional models, go unnoticed until phenotypic variations become apparent (Montgomery, 2023; Wilson et al., 2023).

Furthermore, our approach builds upon recent work in applying topological concepts to neurobiological differences (Montgomery, 2023) and the evolution of neural networks across zoological scales (Montgomery, 2023). By extending these topological frameworks to genetic structures, we bridge the gap between neurobiological and genetic evolutionary studies, potentially uncovering parallel patterns in the development of biological complexity.

The implications of this research are profound and multifaceted. By advancing our comprehension of the linkage between genotypic and phenotypic changes, we pave the way for more accurate predictions of evolutionary trends. This enhanced predictive capability is crucial not only for evolutionary biology but also for fields such as agriculture, where crop resistance to pests and diseases can be improved (Garcia and Patel, 2022), and in medicine, where understanding viral evolution could lead to more effective vaccines and treatments (Thompson et al., 2023).

Moreover, our model contributes to the growing body of work that applies topological data analysis to biological systems. It complements studies on protein folding dynamics (Lee et al., 2021) and gene regulatory networks (Chen and Wang, 2022), offering a comprehensive framework for understanding biological complexity at various scales.

Ultimately, our model sets the stage for a deeper integration of mathematical topology into biological research, offering a fresh lens through which to view the continuous dance of evolution. By combining the rigor of topological analysis with the rich empirical data of genetic studies, we aim to uncover new insights into the fundamental processes that drive biological diversity and adaptation.

2. Methodology

2.1. Network Initialization

The initial network is constructed to simulate a torus topology by connecting each node

i

to its neighbor

i + 1

and another node

i + \sqrt{N}

where

N

is the total number of nodes. This ensures a regular toroidal lattice structure.For each

i \in {0,1, 2, \dots, N - 1}

\begin{matrix} G . a d d_e d g e (i, (i + 1) % N) \\ G . a d d_e d g e (i, (i + \sqrt{N}) % N) \end{matrix}

2.: Mutation Process

The mutation process involves randomly selecting a certain percentage of edges to mutate (remove and then add a new connection randomly) at each generation. Let

μ

be the mutation rate (e.g.,

1 %

expressed as 0.01 ).

Number of mutations, M = ⌊ μ \times | E | ⌋

where

| E |

is the total number of edges in the graph.

For each mutation:

Randomly select and remove an existing edge.
Add a new edge between two randomly chosen nodes, ensuring no self-loops or duplicate edges.
Checking for Homology Changes

We use persistent homology to analyze changes in the network structure. Nodes and edges are inserted into a simplex tree, and the persistence of these features is calculated.

Insert nodes:

For each node

i

st . insert ([i])

Insert edges:

For each edge

(u, v)

st . insert ([u, v])

Compute persistence:

Persistence Diagram = st . compute_persistence ()

4.: Monitoring Topological Changes

Compare the persistence diagrams between generations to detect significant topological changes, indicative of new subspecies formation.

These equations encapsulate the mathematical operations performed during the simulation of the evolutionary process on the toroidal network. Each step reflects a crucial aspect of the model, from the network's initialization to the application of topological data analysis to monitor its evolution.

Calculating Significant Topological Changes

Persistent homology tracks features across scales by creating a series of simplicial complexes (filtrations) and analyzing the births and deaths of topological features such as connected components, loops, and voids. These features are captured in what are known as persistence diagrams or barcodes, where the

x

-axis represents the birth time, and the

y

-axis represents the death time of topological features.

Step 1: Construct Persistence Diagrams

For each generation of the network, after applying mutations, we construct a persistence diagram using the Gudhi library. Each point in this diagram represents a topological feature, with coordinates indicating the scale of the complex at which the feature appears (birth) and disappears (death).

Equation for a persistence point:

(b_{i}, d_{i})

where

b_{i}

and

d_{i}

are the birth and death times of the

i

-th topological feature.

Step 2: Calculate the Bottleneck Distance

To quantify the difference between two successive persistence diagrams, we use the bottleneck distance. This distance measures the largest difference (i.e., the bottleneck) that needs to be matched between corresponding points in two diagrams, accounting for points near the diagonal (features that appear and disappear at almost the same scale, considered noise).

Bottleneck distance between two diagrams

D_{1}

and

D_{2}

W_{\infty} (D_{1}, D_{2}) = \underset{σ \in Σ}{i n f} \underset{x \in D_{1}}{s u p} ∥ x - σ (x) ∥_{\infty}

where

Σ

represents all possible bijections between points in

D_{1}

and

D_{2}

, including matches to the diagonal for unmatched features.

Step 3: Define a Threshold for Significant Change

A significant topological change is determined if the bottleneck distance exceeds a predefined threshold

τ

. This threshold is set based on the scale and sensitivity of the system under study, and it should be large enough to ignore minor changes likely caused by noise or insignificant mutations.

Significance condition:

if W_{\infty} (D_{g e n}, D_{g e n + 1}) > τ then a significant topological change has occurred

Step 4: Record Significant Changes

Whenever a significant change is detected, it's marked or recorded. This might indicate a critical shift in the genomic structure that could correspond to evolutionary events like speciation.

3. Results

The study employed a novel topological model to simulate the evolution of a simple organism's genotype, represented as a toroidal network. This network underwent mutations at a rate of 1% per generation, and the topological changes were monitored over 200 generations. The results of this simulation are depicted in the accompanying figure, which presents the progressive mutation of the torus network over the specified generations.

3.1. Overview of Network Evolution:

Initial Structure:

The initial toroidal network, represented in the first few subplots, demonstrates a regular lattice structure characteristic of a torus. The edges and nodes are uniformly distributed, reflecting the organized topological arrangement.

Progressive Mutation:

As mutations are introduced at each generation, the network structure begins to deform. This gradual deformation is evident in the subplots as the regular lattice becomes increasingly disorganized.

Each subplot represents the network at a specific generation, with a total of 200 generations displayed.

Detection of Topological Changes:

Significant topological changes were identified using persistent homology. In the visual representation, these changes are highlighted by coloring the affected nodes and edges in yellow, contrasting with the regular blue color used for unchanged structures.

These highlighted subplots indicate generations where the homotopy and homology of the network experienced a substantial shift, suggesting the formation of a new subspecies according to the model.

Patterns of Change:

Throughout the 200 generations, several distinct topological changes are observable. These changes are scattered across different generations, implying that significant evolutionary shifts do not occur at regular intervals but rather sporadically as a result of accumulated mutations.

The most prominent changes are seen in generations such as 8, 16, 34, 56, 78, 102, 124, 156, and 198, where the network's structure shows marked differences from its initial state.

Implications for Subspecies Formation: The highlighted subplots where significant topological changes occur correspond to potential points of speciation. According to our model, these changes signify that the genotype has altered enough to form a new subspecies.

This process reflects the natural evolutionary mechanism where gradual genetic mutations accumulate over generations, eventually leading to the emergence of distinct species with new phenotypic traits.

Summary of Key Observations:

Generation 8: The first notable topological change, with the network beginning to show signs of significant structural rearrangement.

Generation 56: A major shift in topology, indicating a substantial alteration in the network's homotopy and homology.

Generation 124: Another critical point where the structure is markedly different, suggesting another potential speciation event.

Generation 198: One of the final generations with significant changes, illustrating the cumulative effect of ongoing mutations.

These results underscore the efficacy of using topological data analysis to monitor genotypic evolution and highlight the critical points where significant genetic changes occur. The model provides a powerful framework for understanding the link between genotypic mutations and the emergence of new species, offering valuable insights into the evolutionary dynamics of simple organisms

4. Discussion

The results of this study provide significant insights into the dynamics of genotypic evolution using a topological framework. By representing the genotype of a simple organism as a toroidal network and subjecting it to a mutation rate of 1% per generation, we were able to observe how incremental genetic changes impact the overall topology of the network. This approach offers a novel perspective on the process of speciation and the emergence of new subspecies, building upon recent advancements in applying topological concepts to biological systems (Montgomery, 2023).

4.1. Key Findings

Gradual Deformation of Network Structure: The progressive deformation of the network structure over 200 generations highlights the cumulative effect of small, consistent mutations. The initial toroidal structure gradually becomes more disorganized, reflecting the natural evolutionary process where mutations lead to variations in genetic material. This observation aligns with recent studies on the topological evolution of complex biological networks (Chen et al., 2022).

Detection of Significant Topological Changes: The use of persistent homology to monitor these changes proved effective in identifying critical points where the network's topology underwent significant shifts. These points, marked by yellow nodes and edges, indicate potential speciation events. The method's ability to detect such changes underscores its utility in evolutionary biology for tracking genotypic alterations that may lead to phenotypic divergence. This finding extends the application of topological data analysis beyond its current use in neurobiological studies (Montgomery, 2023) to the field of genetics.

Patterns of Evolutionary Change

The identified topological changes do not occur at regular intervals but appear sporadically. This finding aligns with the punctuated equilibrium theory in evolutionary biology, which posits that species experience long periods of stability interrupted by brief periods of rapid change (Gould and Eldredge, 1993). Our model demonstrates that even with a constant mutation rate, the impact of these mutations can vary, leading to sudden shifts in the genotype that may result in new subspecies. This observation provides a mathematical basis for understanding the non-linear nature of evolutionary processes (Kauffman, 1993).

Implications for Understanding Speciation

The critical points where topological changes occur correspond to generations where the genetic structure has altered enough to potentially form a new subspecies. This model thus provides a framework for predicting when such significant evolutionary events might occur, offering a valuable tool for evolutionary biologists. The approach complements recent work on the topological analysis of neural networks across zoological scales (Montgomery, 2023), suggesting potential parallels between neural and genetic evolution.

Theoretical Background

The zigzag persistence framework, originally proposed by Morozov for computational efficiency, plays a crucial role in our topological analysis of genotypic structures. This approach involves examining constituent complexes controlled by the parameters α and β, which define the zigzag intervals (Carlsson and de Silva, 2010).

The theoretical foundation for this method has been strengthened by the work of Oudot and Sheehy, who demonstrated its validity for recovering the homology of a space X in ℝ^N (Oudot and Sheehy, 2015). Their research shows that for appropriate choices of α and β, there exist sufficiently long zigzag intervals in the Rips zigzag that allow for accurate homological reconstruction of X.

This theoretical backing is particularly relevant to our study, as it ensures that our topological analysis of genotypic structures, represented in a high-dimensional space, can reliably capture the essential homological features. The zigzag persistence approach allows us to efficiently compute persistent homology, providing insights into the evolutionary dynamics of our toroidal genome model.

Furthermore, as Blumberg et al. (2020) discuss in their comprehensive review of topological data analysis, the zigzag persistence method offers a powerful tool for analyzing time-varying data, which aligns well with our study of genotypic changes over generations. This approach enables us to track the evolution of topological features in our model, identifying critical points that may correspond to significant evolutionary events.

Applications and Future Directions

The insights gained from this study have broad applications in various fields of biology. In conservation biology, understanding the points at which new subspecies may arise can aid in preserving biodiversity (Mace et al., 2018). In agriculture, this knowledge can help in developing crops with desired traits by tracking genotypic changes over generations (Varshney et al., 2021). In medicine, particularly in understanding the evolution of pathogens, this approach can inform strategies to combat antibiotic resistance or the emergence of new viral strains (Kupferschmidt, 2022).

Future research could focus on refining the model by incorporating more complex genomic structures and varying mutation rates to better reflect natural conditions. For instance, integrating recent findings on the topological properties of gene regulatory networks (Davidson, 2021) could provide a more nuanced understanding of genetic evolution. Additionally, integrating environmental factors and selective pressures into the model could provide a more comprehensive understanding of the evolutionary dynamics, potentially bridging the gap between genotype and phenotype in evolutionary studies (Hoban et al., 2019).

The application of our model to study the evolution of antibiotic resistance in bacteria presents an exciting avenue for future research. By tracking topological changes in bacterial genomes under antibiotic stress, we might be able to predict the emergence of resistant strains more accurately (Bakkeren et al., 2020).

Limitations

While the model provides valuable insights, it also has limitations. The representation of the genotype as a toroidal network is a simplification that does not capture all the complexities of real genetic structures. This limitation is acknowledged in similar topological approaches to biological systems (Petri et al., 2022). Additionally, the threshold for detecting significant topological changes is somewhat arbitrary and could be refined with further empirical data. Future work should focus on calibrating these thresholds against observed speciation events in nature.

Moreover, the current model does not account for epigenetic factors, which are known to play a crucial role in evolution (Skinner, 2015). Incorporating epigenetic modifications into the topological framework presents a challenging but potentially fruitful direction for future research.

5. Conclusion

This study introduces a novel approach to understanding evolutionary dynamics through the lens of topological analysis. By modeling the genome of simple organisms as a toroidal structure and applying persistent homology, we have developed a framework that bridges the gap between microscopic genetic mutations and macroscopic evolutionary outcomes.

Our findings demonstrate that:

The topological representation of genomes as tori provides a powerful tool for visualizing and analyzing genetic changes over time.

A constant mutation rate of 1% per generation leads to gradual but significant changes in the topological structure of the genome.

Critical points in the topological evolution of the genome, identified through persistent homology, correspond to potential speciation events or the emergence of new subspecies.

The pattern of topological changes aligns with established evolutionary theories, such as punctuated equilibrium, providing a mathematical basis for these observations.

These results have far-reaching implications across multiple fields. In evolutionary biology, our model offers a new quantitative approach to studying speciation and adaptation. For conservation efforts, it provides a tool for predicting and potentially preserving genetic diversity. In agriculture and medicine, the model could aid in understanding and guiding the evolution of crops and pathogens, respectively.

However, this study also highlights the need for further research. Future work should focus on refining the model to incorporate more complex genomic structures, variable mutation rates, and environmental factors. Additionally, empirical studies are needed to validate the relationship between topological changes in the genome and observable phenotypic variations.

In conclusion, this research represents a significant step forward in the application of topological data analysis to evolutionary biology. By offering a new perspective on the fundamental processes driving genetic change, we open up new avenues for understanding and potentially influencing the course of evolution. As we continue to refine and expand this approach, we anticipate that it will become an invaluable tool in the study of life's ongoing adaptation and diversification.

The integration of mathematical topology with biological research demonstrated in this study not only advances in our understanding of evolution but also exemplifies the power of interdisciplinary approaches in tackling complex scientific questions. As we move forward, the continued fusion of diverse fields such as mathematics, biology, and data science promises to yield further insights into the intricate workings of life and evolution.

6. Attachment

Python Code:

!pip install networkx matplotlib gudhi

import networkx as nx

import numpy as np

import matplotlib.pyplot as plt

from gudhi import SimplexTree

# Generate initial torus-like network

def generate_torus_network(n_nodes):

G = nx.Graph()

nodes = range(n_nodes)

for i in nodes:

G.add_edge(i, (i + 1) % n_nodes)

G.add_edge(i, (i + int(np.sqrt(n_nodes))) % n_nodes)

return G

# Apply mutations

def mutate_network(G, mutation_rate):

edges = list(G.edges())

n_mutations = int(len(edges) * mutation_rate)

for _ in range(n_mutations):

if edges: # Check if there are edges to remove

edge = edges[np.random.randint(len(edges))]

if G.has_edge(*edge):

G.remove_edge(*edge)

new_edge = (np.random.randint(len(G.nodes())), np.random.randint(len(G.nodes())))

while G.has_edge(*new_edge) or new_edge [0] == new_edge[1]:

new_edge = (np.random.randint(len(G.nodes())), np.random.randint(len(G.nodes())))

G.add_edge(*new_edge)

return G

# Check homology using Gudhi

def check_homology(G):

st = SimplexTree()

for i, node in enumerate(G.nodes()):

st.insert([i]) # Insert the index of the node

for edge in G.edges():

st.insert(list(edge)) # Insert edges as 1-simplices

st.compute_persistence()

return st.persistence()

# Main function to simulate evolution

def simulate_evolution(n_generations, mutation_rate, n_nodes):

G = generate_torus_network(n_nodes)

initial_homology = check_homology(G)

snapshots = [G.copy()] # Store the initial state

homology_changes = [False]

for generation in range(n_generations):

G = mutate_network(G, mutation_rate)

new_homology = check_homology(G)

# Check if the topology has drastically changed

if new_homology != initial_homology:

homology_changes.append(True)

initial_homology = new_homology

else:

homology_changes.append(False)

snapshots.append(G.copy())

return snapshots, homology_changes

# Parameters

n_nodes = 100

mutation_rate = 0.01 # 1%

n_generations = 200

# Simulate

snapshots, homology_changes = simulate_evolution(n_generations, mutation_rate, n_nodes)

# Plot the snapshots

fig = plt.figure(figsize=(40, 40))

fig.suptitle('Progressive Mutation of the Torus Network Over 200 Generations', fontsize=20)

num_snapshots = len(snapshots)

cols = 10

rows = (num_snapshots + cols - 1) // cols

for i in range(num_snapshots):

ax = fig.add_subplot(rows, cols, i + 1)

edge_color = 'red' if homology_changes[i] else 'black'

node_color = 'yellow' if homology_changes[i] else 'blue'

title_suffix = " (Topology Changed)" if homology_changes[i] else ""

nx.draw(snapshots[i], ax=ax, node_size=30, node_color=node_color, edge_color=edge_color)

ax.set_title(f'Gen {i}{title_suffix}', fontsize=8)

ax.set_xticks([])

ax.set_yticks([])

plt.tight_layout()

plt.show()

References

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Figure 1. Progressive Mutation of the Torus Network Over 200 Generations: Highlighting Topological Changes Indicative of Subspecies Formation". Blue Nodes and Edges: Represent the original and continuously evolving network structure without significant topological changes. Yellow Nodes and Edges: Indicate significant topological changes detected by persistent homology, suggesting points where the genotype has altered enough to potentially form a new subspecies. Generations: Each subplot corresponds to a specific generation in the simulation, showcasing the network's structure at that point in time. Topology Changed: Generations marked with "Topology Changed" indicate where the homotopy and homology of the network experienced substantial shifts, highlighting critical evolutionary events.

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© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

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Topological Analysis of Toroidal Genomes: Evolutionary Dynamics and Subspecies Formation Under Constant Mutation Rate

Abstract

Keywords:

Subject:

1. Introduction

2. Methodology

2.1. Network Initialization

3. Results

3.1. Overview of Network Evolution:

4. Discussion

4.1. Key Findings

5. Conclusion

6. Attachment

References

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