Dynamics of karyotype evolution

Abstract

In the evolution of species, the karyotype changes with a timescale of tens to hundreds of thousand years. In the development of cancer, the karyotype often is modified in cancerous cells over the lifetime of an individual. Characterizing these changes and understanding the mechanisms leading to them has been of interest in a broad range of disciplines including evolution, cytogenetics, and cancer genetics. A central issue relates to the relative roles of random vs deterministic mechanisms in shaping the changes. Although it is possible that all changes result from random events followed by selection, many results point to other non-random factors that play a role in karyotype evolution. In cancer, chromosomal instability leads to characteristic changes in the karyotype, in which different individuals with a specific type of cancer display similar changes in karyotype structure over time. Statistical analyses of chromosome lengths in different species indicate that the length distribution of chromosomes is not consistent with models in which the lengths of chromosomes are random or evolve solely by simple random processes. A better understanding of the mechanisms underlying karyotype evolution should enable the development of quantitative theoretical models that combine the random and deterministic processes that can be compared to experimental determinations of the karyotype in diverse settings.

Chromosome structure (the karyotype) is usually considered to be the same in all cells of the body and in all individuals in a given species. However, one of the hallmarks of cancer involves instability of the chromosome structure leading to abnormal numbers and types of chromosomes in cancerous cells. Likewise, during the course of evolution, the numbers and sizes of chromosomes change so that even closely related species can have different numbers of chromosomes. Emerging technologies enabling large scale determination of DNA sequences are exponentially increasing our knowledge about the types of changes and variation in chromosome organization that occur in cancerous tissue in a single individual, in different individuals of a species, and in different species. We suggest that understanding the mechanisms of the dynamics of karyotype evolution will be facilitated by the consideration of karyotype evolution from a nonlinear dynamics perspective.

“I believe there is little reason to question the presence of innate systems that are able to restructure a genome. It is now necessary to learn about these systems and to determine why many of them are quiescent and remain so over very long periods of time only to be triggered into action by forms of stress, the consequences of which vary according to the nature of the challenge to be met. In addition to modifying gene action, these elements can restructure the genome at various levels, from small changes, involving a few nucleotides, to gross modifications, involving large segments of chromosomes, duplications, deficiencies, inversions, and other more complex reorganizations.”

Barbara McClintock (1978)¹

“Genome change does not occur accidentally.”

James A. Shapiro (2023)²

I. INTRODUCTION

The karyotype refers to the characteristics of the chromosomes. Such characteristics include the number of chromosomes, the size of chromosomes, locations of centromeres (the points where two chromosomes bind during cell division), and banding patterns of chromosomes under the presence of various stains that bind to specific chromosomal regions. These characteristics can be most easily observed in cells during the process of cell division. In fertilization, one set of chromosomes comes from the male and a second set from the female. Consequently, the number of chromosomes in a species is usually an even number, so that each chromosome is paired with another. Although all organisms of a species have the same karyotype, advances in molecular biological techniques have made it clear that there are a large number of structural variants in the genome in which segments of chromosomes may differ from one individual to the next.^3,4 When two close species with different numbers of chromosomes are bred, for example, a horse (2n=64) and a donkey (2n=62), the resulting hybrid animal (here a mule) is generally not able to further reproduce; it is not fertile. The observation of constancy of karyotype in a species and infertility of hybrid progeny from mating of animals with different karyotypes pose a dilemma:“How can new species with a different karyotype emerge?” Research in genomics and cytogenetics addresses this question.^4–9

A related issue involves variation of the genome organization and karyotype in a single organism. Although all cells are derived from the fertilized egg, during the course of development different cells in an organism can develop differences in chromosome structure.¹⁰ A striking example occurs in cancer: a large number of cancers are associated with chromosomal reorganization.^11–16 Further, many particular types of cancer are associated with similar changes in the karyotype.¹⁷ Consequently, understanding the mechanics and modifications of chromosome dynamics can have important implications for health.^13,17,18

The premise of the current article is that the “innate systems that are able to restructure a genome” can be thought of as a dynamical system. In a given organism and in a given species the karyotype is a fixed point—it usually does not change over time. However, changes in the karyotype do occur, whether in the emergence of a new karyotype associated with a new species or the establishment of a clone of cells with an altered karyotype that may lead to cancer. A vast amount of research has been directed at documenting changes of karyotype that occur and identifying mechanisms that lead to these modifications in speciation and carcinogenesis. Although most of this work is either related to just speciation or just carcinogenesis, some research points out common problems in both areas.^19–21 An underlying question is whether changes that occur arise solely by random processes followed by selection, or whether some aspects of the changes reflect a combination of random and deterministic processes. In this article, we describe dynamic aspects of structural variations of chromosomes and the karyotype. This article is in honor of David Campbell’s 80th birthday. As the founding editor of Chaos, David’s leadership, creativity and integrity have helped shape the development of the field of nonlinear dynamics.

In Sec. II, we briefly review some basic properties of chromosomal structure and nonlinear dynamics. In Sec. III, we discuss mechanisms for karyotype changes between cells in a single individual with emphasis on karyotypic changes in cancer. Section IV deals with the karyotype in evolution of species. We show that mathematical models that have been proposed for the evolution of the karyotype can be thought of from a dynamical systems perspective. We review statistical analyses of gross features of the karyotype. These types of analysis can give information about the rates of genome rearrangements in evolution. A common conclusion of the analysis and modeling is that purely random models do not agree with the observed features of karyotypes. Section V discusses future directions.

II. BASICS CONCERNING CHROMOSOME STRUCTURE AND NONLINEAR DYNAMICS

We first present some basic facts about chromosomes and cell division. We consider an organism with $2n$ chromosomes. Mitosis is the division of a somatic cell to give two daughter cells. Meiosis is the process by which a stem cell gives rise to sperm cells and ova. During mitosis, chromosomes duplicate and a complete set of chromosomes goes to each daughter cell. During this process, each pair of chromosomes is attached to a cell spindle in a specific region called the centromere. If the centromere is near the middle of the chromosome, the chromosome is metacentric. If the centromere is near the end of the chromosome, the chromosome is acrocentric. The segments of the chromosome on either side of the centromere are called arms, and by convention the shorter arm is designated p and the longer arm is designated q. Thus, in humans, 4p refers to the shorter arm of chromosome 4. In meiosis, there is one duplication of chromosomes but two cell divisions. During the first cell division, there is crossing over so that parts of a chromosome pair from the maternal and paternal chromosomes are interchanged. In meiosis leading to sperm cells, one stem cell leads to four sperm cells, where each sperm cell has one complete set of $n$ chromosomes. In the process leading to ova, only one ovum is generated from each germ cell. The other three cells are called polar bodies. Normal somatic cells are diploid with two complete sets of chromosomes, whereas sperm and ova are haploid and only have one complete set of chromosomes. Cells with an abnormal number of chromosomes are aneuploid.

Chromosomal structural modification occurs via several processes some of which leave the number of chromosomes constant, whereas others change the number of chromosomes. Mechanisms that leave the number of chromosomes constant include (i) deletion in which a segment of a chromosome is deleted; (ii) inversion in which a given segment of chromosome becomes inverted; (iii) translocation in which two segments of different chromosomes exchange places, perhaps with inversion of one or both segments; and (iv) duplication in which a segment of a chromosome is duplicated. The number of base pairs involved in any of these processes can vary greatly. When the above processes are confined to relatively small segments of the chromosome, they lead to structural variants.^3,4 Processes that change the number of chromosomes include (i) fusion in which two chromosomes fuse, often occurring when two acrocentric chromosomes fuse; (ii) fission, in which a single chromosome breaks into two, often at the centromere; and (iii) duplication in which one or more chromosomes undergoes duplication.

Nonlinear dynamics deals with how things change over time. We call $K(t)$ the karyotype at time $t$, $K_t$ is governed by an equation

$$K_{t+1}=f(K_t),$$ (1)

where $f$ represents the rules for karyotype evolution. Depending on the particular context, the karyotype might reflect different levels of fineness of description. For example, as we will discuss in Sec. IV, the karyotype might simply be constituted by the number and lengths of each of the chromosomes without considering any of the finer structures. The function $f$ might contain both deterministic and stochastic operations. The function $f$ does not necessarily have an algebraic representation but can be composed of some set of rules for altering the geometry of the karyotype. The meaning of $t$ depends on the context. When considering cancer, $K_t$ is the karyotype of a mother cell and $K_{t+1}$ is the karyotype of the daughter cells. When considering evolution of new species, $K_t$ could represent the karyotype of an individual and $K_{t+1}$ the karyotype of a descendant or of a newly evolved species. If the karyotype does not change, then $K_{t+1}=K_t$.

III. KARYOTYPE VARIATION IN CELLS IN AN ORGANISM

Many of our genes are involved in carrying out and controlling key cellular functions including progression through the cell cycle, cell division, and repair of mistakes made while copying DNA. Especially important from the context of cancer are tumor suppressor genes and oncogenes. TP53 is a tumor suppressor gene that encodes a protein that is a master transcription factor regulator of the surveillance system that forces genetically damaged cells to undergo cell death. If TP53 is inactivated, defective cells can propagate resulting in cancerous growth.²² Other well-studied tumor suppressor genes are BRCA1 and BRCA2, which encode proteins that function in a DNA repair mechanism by homologous recombination in which the DNA sequence in an undamaged sister chromatid is copied in order to repair DNA double-strand breaks that happen during DNA replication. MYC is an oncogene that regulates the transcription of hundreds of genes contributing to a range of processes including proliferation, cell survival, and genomic instability.²³ Mutations that lead to a loss of function of tumor suppressor genes, or modified or enhanced activity of oncogenes leads to a situation called genome instability characterized by a rapid rate of mutations. The accumulation of these genomic changes has been called microevolution²¹ and often is a key step in the development of cancer. Genomic instability is an enabling characteristic and a hallmark of cancer.^24,25

In contrast to the comparatively localized microevolutionary changes, large aberrations of chromosomal structure also commonly occur in cancer. This was recognized over a hundred years ago by von Hansemann and Boveri who observed that cancerous cells often display aneuploidy.^11,12 Such karyotypic changes reflect macroevolutionary processes that shape cancer genomes.²¹ In describing chromosome changes leading to aneuploidy in cancer, researchers identified a process called chromosomal instability, in which whole chromosomes or parts of chromosomes are gained or lost.²⁶ Chromosomal instability occurs as a result of failure of chromosomes to properly attach to or migrate along the spindle assembly that is established during mitosis and directs chromosomes to the two daughter cells.^26,27 Chromosome size and gene density correlate with aneuploidy; the larger the chromosome and the lower the gene density, the higher the rate of mis-segregation.²⁸ Large chromosomes tend to be positioned farther away from the nucleus center, which increases their likelihood for mis-segregation.²⁸ The specific chromosome locations in nuclei vary across tissues further influencing the aneuploid patterns.²⁹ Chromothripsis is another process that generates chromosome abnormalities leading to massive shattering of mis-segregated chromosomes.³⁰ Gains, losses, and rearrangements that result from DNA damage repair of the shattered chromosomes may vary due to differences in the activity of DNA double-strand repair pathways active in particular tissue or tumor types.³¹ Additionally, homologous recombination of repetitive sequences in the genome may play a role in the resulting gains and losses since it has been reported that the breakpoints in cancer genomes are statistically enriched at the sites of structural variants in the human population.³² Since a subset of all genes in the genome is expressed in any given tissue, the balance of the expressed tumor suppressors and oncogenes drives the selection for specific aneuploid events.^33,34 Since all these processes occur at a molecular level, there are necessarily stochastic aspects that interact with the deterministic mechanisms of cell division and repair.

Genome-wide studies showed that chromosome arm level aberrations are more common than whole chromosome aberrations and certain chromosomal arms are preferentially lost or gained, suggesting that these events are selected because they are advantageous for tumor fitness.^36–38 Chromosome arm aberrations recur within individuals of the same cancer type and across cancer types such as breast cancer and stomach cancer (Fig. 1). For example, a common alteration is chr17p deletion, which harbors TP53 and is thought to lead to its biallelic in activation in cases which also exhibit loss of function coding mutations in TP53. Chromosome arm losses are more frequent and occur earlier than gains in cancer (Fig. 1). Cancer type specific aneuploidy is also observed such as glioblastoma and reflect tissue specific gene expression differences that influence chromosomal arrangement in the nuclei and expressed tumor suppressor genes and oncogenes. The gene dosage balance hypothesis³³ proposed that chromosome arm loss will be favored if the number of tumor suppressor genes is higher than oncogenes and vice versa in the case of gain and conversely chromosome arm gain will be favored if the number of oncogenes is greater than the number of tumor suppressor genes. This is supported by findings that in normal cells the overexpression of consecutive genes within specific chromosomal regions that reduces cell growth correlates with regions of chromosome arm loss in cancers, whereas regions of chromosomes associated with increased cell growth tend to be gained.³⁴

FIG. 1.

The landscape of large chromosomal aberrations in cancer. (a) Evolutionary timing results for chromosome arm aberrations as reported for BRCA, STAD, and GBM¹⁵ and for BRCAbasal.³⁵ Early and late clonal phases are timed with respect to whole genome duplication (WGD). Dashed line separates the p and q chromosome arms. Each box is filled with a color according to its evolutionary timing (light green for early clonal, light purple for late clonal), shape reflects a gain (circle) or a loss (square). (b) The frequency of observed gains, losses, and whole genome duplication (WGD) across different chromosome arms in a cohort of primary (TCGA) and metastatic¹⁶ tumors, separated by tumor type. BRCA, breast invasive carcinoma; STAD, Stomach Adenocarcinoma; GBM, glioblastoma multiforme; BRCAbasal, basal subtype of breast invasive carcinoma. The molecular subtype of basal breast cancer is enriched for a histological subtype of triple negative breast cancer. Dashed line separates the p and q chromosome arms. Each box is filled with a color according to its copy number: a gain is in red and a loss is in blue. Primary tumors are shown in a light shade and metastatic tumors are shown in a dark shade.

These insights are shaping the current research of co-authors Kuzmin, Baker, and Van Loo who are analyzing chromosome evolution in triple negative breast cancer³⁵ and other cancers more broadly.¹⁵ Triple negative breast cancer, which is a subtype of breast cancer with the worst prognosis, does not express estrogen or progesterone receptors and also does not have an excess of a specific protein, HER2, that promotes tumor growth. However, triple negative breast cancer displays recurrent large chromosomal deletions.^35,39 Observation that certain chromosome arms, such as chromosomes 8p, 5q, and 4p, are frequently deleted in triple negative breast cancer, led to the characterization of selection pressures driving a poorly known chromosome 4p loss.³⁵ It evolves early in tumorigenesis and is functionally significant since it is associated with a proliferative, dedifferentiated and immune evasive state. It also led to the identification of a novel tumor suppressor C4orf19 (PGCKA1). The close proximity of genes with similar function on chromosome 4p identified in triple negative breast cancer,³⁵ as revealed by suppression of proliferation when overexpressed, is consistent with findings in yeast where genes with similar effects on the phenotype are situated closer to each other in the genome than expected by chance alone.⁴⁰

A key focus of current research is identifying the evolutionary history of the genome in different types of cancer. A pan-cancer whole genome sequencing analysis across multiple cancers and a more focused analysis on triple negative breast cancer revealed that coding mutations in TP53 and chromosomal arm losses emerge early in tumor evolution followed by whole genome doubling and chromosomal arm gains,^15,35 Fig. 1. In some cancers, mutations in single genes are observed in later evolutionary stages, such as subclonal stages following a burst of early large chromosomal deletions.¹⁵ The theory of punctuated copy number evolution of cancer proposes that chromosomal instability leads to bursts of chromosomal arm aberrations, enabling co-selection of an advantageous subset of aneuploid events, which are stably clonally propagated.⁴¹ The observation that multiple aneuploid events co-occur indicates that genetic interactions among aneuploidies are important factors in shaping cancer genome evolution. This was supported by an observation that triple negative breast cancer cell lines deleted for chromosome 4p exhibited stronger suppression of proliferation than their normal counterparts suggesting selection for co-occurrence.³⁵ This is also consistent with experiments in yeast which demonstrated genetic interactions between aneuploidies.⁴²

Due to the difficulties in carrying out studies of genome evolution of cancers in humans, a number of model systems are being developed. A transgenic mouse model of triple negative breast cancer with a deletion of TP53 shows a spontaneous loss of a region on chromosome 11, which corresponds (is syntenic) to a region on human chromosome 5q.¹⁸ Since chromosome 5q is often lost in triple negative breast cancer in humans, there is an indication that large chromosomal aberrations associated with cancer are conserved across species.

Other examples involve experiments in tissue culture. An experimental model used CRISPR-Cas9 methodology to delete the tumor suppressor gene TP53 in organoid cultures of normal gastric tissue organoids.⁴³ The organoids were followed for over two years. Aneuploidy occurred in a similar preferred sequential order as observed in triple negative breast cancer suggesting convergent mechanisms for regulating tumor emergence.^15,35,43 Finally, cell culture models established from different tissue types with engineered aneuploidies show that aneuploid genomes accumulate additional chromosomal gains and losses to evolve an improved fitness state.^44–46 These findings suggest an important role of genetic interactions in the evolution of complex aneuploid patterns observed in cancer.

In summary, current research is delineating trajectories for genomic and chromosomal evolution in cancer. Patients with similar types of cancer often show similar trajectories. Some cancers, e.g., breast cancer and gastric cancer, have similar trajectories, whereas other cancers such as glioblastoma have different trajectories, Fig. 1. Thus, nonrandom macroevolutionary processes appear to operate to shape cancer genomes. More nuanced in silico modeling may be useful to gain a deeper understanding of cancer evolution and ultimately develop effective cancer therapies.

IV. DYNAMICAL MODELS FOR SPECIATION AND KARYOTYPE EVOLUTION

A. Random mechanisms for generating chromosome lengths

Segments of chromosomes are rearranged over the course of evolution of new species. Within a given segment, genes might undergo mutations of individual base pairs, but in any given segment the same genes maintain their proximity. As genetic and molecular biological techniques have become more refined, our understanding of the nature of these rearrangements has changed.

In early work, Nadeau and Taylor carried out an analysis of the lengths of chromosomal segments conserved in evolution in man and mouse.⁴⁷ They concluded that the observed exponential distribution of segment lengths, in which short segment lengths were more common than long lengths, was consistent with a random process for genome rearrangement. This has led to theoretical models for chromosomal evolution in which concepts of random rearrangement are made precise.^48,49

To analyze chromosome lengths, it is useful to normalize the total lengths of the chromosomes to 1. If the chromosomes are now laid end to end, we can define a series of points $x_1,x_2,\dots ,x_{n-1}$ in the interval $(0,1)$ and take $x_0=0$ and $x_n=1$. The distance $y_j=x_j-x_{j-1}$, $1\le j\le n$ corresponds to the normalized length of the $j$th chromosome.

The distribution of lengths of chromosomes can be compared to theoretical distributions generated by some specific mechanism. Suppose, for example, that chromosomal evolution is generated by random processes. Then, it might be plausible that the probability density of chromosome lengths for an organism with 2 $n$ chromosomes would be comparable to the length distribution of line segments generated by a random selection of $(n-1)$ points in the interval $(0,1)$. For this random process, the probability density of lengths between $y$ and $y+\mathrm{\Delta }y$ is $p(y)\mathrm{\Delta }y$, where⁵⁰

$$p(y)=(n-2){(1-y)}^{(n-1)}.$$ (2)

In humans, there are 46 chromosomes where 22 pairs are autosomes and 1 pair is the sex chromosomes. Recall that males have an X and Y chromosome which are different lengths in humans, and females have two X chromosomes.⁵¹ To compare the human chromosome distribution with a random length theoretical model, we only consider the 22 autosomes. In Fig. 2, we show the density distribution for humans, green bars. We compare this with the theoretical probability density function from Eq. (2), black curve. In addition, we have generated 1000 random sets of 22 model chromosomes with random length and determined the resulting probability density distribution, blue bars. This randomly generated length distribution agrees with the theoretical density distribution. However, as observed in earlier work,^48,49 for the human genome, as well as the genomes in a large range of other species, the chromosome lengths are more clustered at intermediate values. There are fewer long and short chromosomes than expected in a purely random model for chromosome lengths.

FIG. 2.

Histogram of chromosome sizes comparing Eq. (2) (black line), random simulation (blue), and humans (green). The percentage of chromosomes expected in each bin is given as a function of the fraction of the total chromosome length for each chromosome. The blue bars are based on 1000 randomly generated sets of 22 chromosomes and the green is the actual distribution of human chromosome lengths.⁵¹

An extension of the random length model is to translocate randomly selected segments from two chromosomes, where the probability of choosing a chromosome for random exchange is proportional to its length.^48,49 We have carried out simulations for a system with 22 chromosomes starting from an initial condition in which all lengths are equal. In doing the simulation, we choose two points for translocation randomly so that the probability of a translocation locus falling on a chromosome is proportional to the chromosome length. If the two points fall on the same chromosome this does not change the length distribution, but still counts as one iteration. We can think of this iterative process as an example of Eq. (1), where the function is a random translocation. In Fig. 3(a), we show the distribution after 100 translocations and in Fig. 3(b) we show the distribution after 1500 translocations. The distribution has not converged after 100 translocations, but has converged to the random model given by Eq. (2) by 1500 iterations. This agrees with earlier observations,⁴⁸ but to the best of our knowledge is not proven.

FIG. 3.

Histograms of chromosome sizes starting with 22 equally sized chromosomes and carrying out random translocations. The percentage of chromosomes expected in each bin is given as a function of the fraction of the total chromosome length. The blue bars are averages based on 1000 trials starting from the same equal length distribution. Panel A gives the result following 100 translocations and panel B gives the result after 1500 translocations. The black curve is the distribution of lengths for a line divided by 21 randomly chosen points [Eq. (2)].

One attempt to modify the random translocation model involves putting lower and upper limits on the chromosome lengths. Thus, any random translocation that would lead to a chromosome too long or too short would not be allowed. This modification gives better agreement with the observed distributions of chromosome lengths over a broad range of species.^49,52 Another possibility would be to allow translocations that lead to chromosomes above a certain length limit, but then to carry out a fission of the units that were too long leading to an increased number of chromosomes. Likewise, if a chromosome was too short, there would be a fusion of the short chromosome with another chromosome in the set. Once the upper and lower limits were set, one would expect that asymptotically one could determine the stable distributions of both the numbers of chromosomes and their size distribution. The availability of databases containing the numbers of chromosomes of different species provides a source of data for comparison with computational models.⁵³

B. Rate of chromosome evolution

Advances in cellular biology and molecular biology have enabled detailed analysis of the ways in which genomes and karyotypes have evolved. Individual bases of DNA molecules are subject to mutation. Since many such mutations are not harmful, as time proceeds the DNA sequences for the same protein in related species can gradually diverge. By estimating this rate, an estimate to the times of establishment of new species can be made. Similarly, as evolution proceeds, the various processes mentioned above that lead to new chromosome geometries can reshuffle segments of chromosomes. A remarkable technique known as chromosome painting is able to directly determine the locations of segments of chromosome that have a common origin (i.e., they are homologous). Such techniques have been applied, for example, to analyze and compare chromosome structure in several dog species,⁵⁴ mice and humans,⁷ and also in several species of deer and related animals.⁵⁵

The study of Mudd et al.⁵⁵ (see Fig. 4) demonstrates the power of these techniques. They compare karyotype in cows (B. taurus), Indian muntjac deer (M. muntjak), Chinese muntjac deer (M. reevesi), red deer (C. elaphus), and reindeer (R. tarandus). Although the Indian and Chinese muntjac deers are believed to have diverged about 4.9 $\times {10}^6$ years ago, their karyotypes are very different. In the Indian muntjac 2n=6, whereas in the Chinese muntjac 2n=46. The rate of karyotype change in the Indian muntjac was estimated to be $5.3$ changes per million years, which is estimated to be an order of magnitude faster than in other mammalian species.⁵⁶ The observation that all the changes are associated with fusion, rather than other karyotype modifications, is consistent with earlier analyses by White, who proposed the term “karyotypic orthoselection” to describe a situation in which there are multiple changes of karyotype of a similar kind between two related species.^6,57 In the present case, by combining analysis of genetic mutations between the related species, Mudd et al. suggested that mutations in genes that code for proteins that are “central in DNA metabolism and chromosome biology $\cdots$ may have contributed to establishing a permissive cellular environment that allowed successive fusion events and the rapid evolution of muntjac karyotypes.” It is also possible that the observed karyotype changes are directed by more active processes that direct and control the karyotypic changes.

FIG. 4.

Chromosome rearrangements from several related species. Panel A gives a timeline for the emergence of new species. Panel B shows how segments of different chromosomes have been reorganized and rearranged in the course of evolution. Panel C gives the evolutionary distance between related species. Reproduced with permission from Mudd et al., Commun. Biol. 3, 480 (2020). Copyright 2020 Author(s), licensed under a Creative Commons Attribution 4.0 License.

V. DISCUSSION

One of the remarkable characteristics of living systems is their ability to accurately copy DNA molecules and chromosomes and to package the resulting structures in viable daughter cells. The process works in germ line cells, generating cells with a haploid number of chromosomes as well as somatic cells where each daughter cell contains a diploid number of chromosomes. The involved processes occur at a molecular level. Despite high levels of accuracy, there are many circumstances in which there is no absolute fidelity in the copying processes.

Some of these processes are programmed into the genetic machinery and are crucial for the proper functioning of the organism. During the development of germ cells, there is crossing over between the maternal and paternal chromosomes so that each germ cell contains DNA segments from both parental cells. In the development of immune cells, certain regions of the DNA are specifically cut and spliced so that the resulting daughter cells can express different antibodies. In these processes, the loci of the cutting and splicing are not always random; there can be specific loci where the cutting is favored in both germ cells⁵⁸ and immune cells.⁵⁹ Modification of the karyotype also plays an important but often unappreciated and unacknowledged role in the evolution of species⁶⁰

In contrast to the situations in which the lack of fidelity of duplication of DNA is essential for normal functioning, errors in chromosome duplication can lead to pathology. There are well-known abnormalities associated with germ cell production, such as trisomy 21 associated with Down syndrome. In somatic cells, genomic and chromosomal modifications play a major and in some cases a defining role in the initiation and progression of cancer.

Understanding the mechanisms underlying the evolution of species or the establishment of cancerous cells is a central focus of research. For karyotypic changes leading to speciation to be passed on, they must lead to improved reproductive ability of individuals. For karyotypic changes leading to cancer, there must be a failure of normal processes that control mitosis and uncontrolled growth of the resulting cell lines. One possibility is that these changes occur as a consequence of random mutations followed by selection. Our main point in this article is that there are many results that are inconsistent with the assumption that karyotypic change occurs solely by random mutations. Specifically, we have discussed the following findings.

• •In evolution of species, there are many instances in which the same type of a change occurs multiple times in the evolution of a given species. Thus, muntjac deer with 2n=6 has evolved from an ancestral species believed to have 2n=58 by a series of fusions of chromosomes in 5 $\times {10}^6$ years.⁵⁵ This tendency toward similar changes in karyotype in the course of evolution was called karyotypic orthoselction by White.⁵
• •Sites in the genome at which chromosome breakage and rearrangement occur do not appear to be random, but have certain structural features that predispose them to chromosome breakage and rearrangement. Some sites of chromosome fragility in cancer have been associated with sites of chromosome breakpoints for transpositions and inversions in evolution of species^8,61,62
• •Cancers affecting single and multiple organs often have similar karyotypes and karyotypic changes evolve with similar timing. For example, different cancer types, such as triple negative breast cancer and stomach cancer, are often marked by similar evolutionary clonal losses, for example, chromosomes 17p and 5q. Similar karyotypes in cancers are thought to evolve due to the differences in chromosome segregation,²⁸ DNA damage repair,³² gene density,²⁸ balance of tumor suppressor genes and oncogenes,³³and genetic interactions among aneuploidies.^35,42 Thus, in addition to the length of chromosomes, the features associated with DNA replication, the biophysical properties of cell division, and the topological relationship among genes along and between chromosomes are important factors in shaping karyotype evolution.
• •Although it is difficult to track the progression of karyotypic changes in cancer in single individuals, there is growing evidence that similar types of changes occur in different individuals. A model system composed of organoids in tissue culture that is an experimental model of the emergence of gastric cancer showed similar changes in replicate organoids that appeared to recapitulate changes that occur in gastric cancers in patients.⁴³
• •Although mathematical modeling of karyotypic evolution has been limited, an early sharp result from Sankoff and Feretti⁴⁸ found that the statistical properties of chromosomal length distribution are inconsistent with a model of random transpositions between chromosomes. In the future, it would be illuminating to characterize chromosome size distribution in cancers and compare its relation to species karyotypes.

The cited quotations from Barbara McClintock and James Shapiro at the start of the article forcefully state that changes in the karyotype do not solely arise by chance, but also reflect innate mechanisms. Is it possible that the evolution of the karyotype is determined solely by deterministic factors and that karyotype evolution is “chaotic” in the technical sense used in nonlinear dynamics? Due to the molecular scale of the subcellular processes involved with chromosome metabolism, stochastic processes seem inevitable in chromosomal duplication and segregation during meiosis and mitosis. However, further investigations of mechanisms that underlie minor and major chromosome reorganization are needed. If such mechanisms were better understood, it should be possible to develop quantitative theoretical models that could be used to quantitatively evaluate the relative roles of stochastic and deterministic factors in karyotype dynamics.

One factor that makes it difficult to develop models for the evolution of the karyotype is the difficulty in acquiring high accuracy quantitative data for karyotype dynamics. All such data necessarily involves determining chromosome features, ranging from the number and size of chromosomes to the DNA sequence over time. Although it is possible to make inferences about genome changes over time from analysis of a snapshot of multiple genomes at a single time, it would be preferable to determine sequential changes in genomes directly at multiple times.⁶³ In the case of cancer progression, new advances in cell and molecular biology are enabling determination of genomic structure at multiple time points and from multiple points in a tumor but such determinations remain practically difficult and expensive to implement even in highly simplified model systems.⁴³

One goal of the current perspectives article is to bring the vast and dispersed literature concerning karyotype dynamics and evolution to the attention of the nonlinear dynamics community. The underlying concept driving this perspective is that in a given species or individual the karyotype is a fixed point and that circumstances that lead to modifications of the karyotype bear some similarities in circumstances that lead to bifurcations that occur in dynamical systems. A similar notion was put forward recently by Heng and Heng²⁰ who stated “Genome behavior during crises shares common features with complex systems described by chaos theory.” One fundamental aspect of nonlinear dynamics is the ability to carry out analytic determinations of stability of fixed points and oscillations based on the underlying dynamical equations. In translating this perspective to karyotype evolution, it is essential to focus on the specific characteristics of the genome and environment that lead to instability. Research has identified a number of factors including local DNA structure and mutations at target sites for chromosome breakage, mutations in proteins involved with identification of sites in which there is a lack of fidelity of DNA replication, mutation in proteins involved with repair of DNA. As well, environmental stresses including some therapies used in cancer such as radiation or chemotherapy facilitate karyotypic evolution.

Fruitful quantitative theoretical work involves reconstructing sequences of chromosomal evolution and change based on careful determinations and comparisons of chromosomal structures from different species or from different clones of cancerous cells from a single individual over time from benign to malignant state. In the future, it is important to construct models that can reproduce statistical properties of size distributions of chromosomes.

We conclude with a note on the current paper. In July 1983, L.G. presented some of these notions about karyotypic evolution and speciation in unpublished notes that included a draft version of Fig. 2. The “very rough notes” were occasionally discussed with colleagues over the ensuing years. In 2017, L.G. attended a conference in Barcelona at which Thomas Ried gave a talk about chromosomal instability in cancer. Discussions between L.G. and Ried led to a renewed interest in karyotypic evolution. A talk by E.K. concerning her current research on chromosomal modifications occurring in breast cancer in Montreal in November 2023 underscored the importance of chromosomal instability in the development of cancer. Further discussions of the ideas between L.G. and E.K. led to the decision to present the current perspectives article. The literature in this area is vast; we have tried to give sufficient references to facilitate entry to the area by outsiders but could not give a comprehensive review. We apologize in advance to any whose work was omitted or not given sufficient emphasis. Given the exponential increase in the ability to sequence and analyze DNA in multiple species and cancerous tissue, it is clear that quantitative analyses of the resulting data are necessary. Data scientists with expertise in genomics will play essential roles. But we believe that a nonlinear dynamics perspective, that seeks out deterministic mechanisms to understand complex seemingly random events, may also be needed. We hope the current work stimulates research in that direction.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Elena Kuzmin: Conceptualization (equal); Data curation (equal); Writing – original draft (lead); Writing – review & editing (equal). Toby M. Baker: Data curation (equal); Writing – review & editing (supporting). Peter Van Loo: Data curation (equal); Writing – review & editing (supporting). Leon Glass: Conceptualization (equal); Formal analysis (equal); Software (equal); Writing – original draft (lead); Writing – review & editing (lead).

DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.