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Wagner N, Atsmon-Raz Y, Ashkenasy G. Theoretical Models of Generalized Quasispecies. Curr Top Microbiol Immunol 2016; 392:141-59. [PMID: 26373410 DOI: 10.1007/82_2015_456] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/11/2023]
Abstract
Theoretical modeling of quasispecies has progressed in several directions. In this chapter, we review the works of Emmanuel Tannenbaum, who, together with Eugene Shakhnovich at Harvard University and later with colleagues and students at Ben-Gurion University in Beersheva, implemented one of the more useful approaches, by progressively setting up various formulations for the quasispecies model and solving them analytically. Our review will focus on these papers that have explored new models, assumed the relevant mathematical approximations, and proceeded to analytically solve for the steady-state solutions and run stochastic simulations . When applicable, these models were related to real-life problems and situations, including changing environments, presence of chemical mutagens, evolution of cancer and tumor cells , mutations in Escherichia coli, stem cells , chromosomal instability (CIN), propagation of antibiotic drug resistance , dynamics of bacteria with plasmids , DNA proofreading mechanisms, and more.
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Affiliation(s)
- Nathaniel Wagner
- Department of Chemistry, Ben-Gurion University of the Negev, Beer Sheva, 84105, Israel
| | - Yoav Atsmon-Raz
- Department of Chemistry, Ben-Gurion University of the Negev, Beer Sheva, 84105, Israel
| | - Gonen Ashkenasy
- Department of Chemistry, Ben-Gurion University of the Negev, Beer Sheva, 84105, Israel.
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Shirmovsky SE, Boyda DL. Study of DNA conducting properties: reversible and irreversible evolution. Biophys Chem 2013; 180-181:95-101. [PMID: 23911953 DOI: 10.1016/j.bpc.2013.06.014] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/25/2013] [Revised: 06/18/2013] [Accepted: 06/18/2013] [Indexed: 11/19/2022]
Abstract
A hole transport through DNA base sequences was modeled. The fact that DNA consists of two polynucleotide strands was taken into consideration. Specific DNA base locations are determined in the model. The model predicts the behavior of a hole acting on the DNA chain, taking into account reversible and irreversible dynamics. It was shown that the transfer mechanisms depend on the sequence type and can be either of hopping nature or of superexchange one. Distance dependence of the hole transport relative rate on the number of hopping steps was investigated. The results obtained were compared with the experimental data. The investigation demonstrates the utilization of the formalism in practical problems for description of the charge migration through the different molecular sequences.
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Affiliation(s)
- S Eh Shirmovsky
- Theoretical and Experimental Physics Cathedra, Far Eastern Federal University, 8 Sukhanov St., Vladivostok 690950, Russia.
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Rübben A, Nordhoff O. A systems approach defining constraints of the genome architecture on lineage selection and evolvability during somatic cancer evolution. Biol Open 2012; 2:49-62. [PMID: 23336076 PMCID: PMC3545268 DOI: 10.1242/bio.20122543] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/17/2012] [Accepted: 10/15/2012] [Indexed: 12/22/2022] Open
Abstract
Most clinically distinguishable malignant tumors are characterized by specific mutations, specific patterns of chromosomal rearrangements and a predominant mechanism of genetic instability but it remains unsolved whether modifications of cancer genomes can be explained solely by mutations and selection through the cancer microenvironment. It has been suggested that internal dynamics of genomic modifications as opposed to the external evolutionary forces have a significant and complex impact on Darwinian species evolution. A similar situation can be expected for somatic cancer evolution as molecular key mechanisms encountered in species evolution also constitute prevalent mutation mechanisms in human cancers. This assumption is developed into a systems approach of carcinogenesis which focuses on possible inner constraints of the genome architecture on lineage selection during somatic cancer evolution. The proposed systems approach can be considered an analogy to the concept of evolvability in species evolution. The principal hypothesis is that permissive or restrictive effects of the genome architecture on lineage selection during somatic cancer evolution exist and have a measurable impact. The systems approach postulates three classes of lineage selection effects of the genome architecture on somatic cancer evolution: i) effects mediated by changes of fitness of cells of cancer lineage, ii) effects mediated by changes of mutation probabilities and iii) effects mediated by changes of gene designation and physical and functional genome redundancy. Physical genome redundancy is the copy number of identical genetic sequences. Functional genome redundancy of a gene or a regulatory element is defined as the number of different genetic elements, regardless of copy number, coding for the same specific biological function within a cancer cell. Complex interactions of the genome architecture on lineage selection may be expected when modifications of the genome architecture have multiple and possibly opposed effects which manifest themselves at disparate times and progression stages. Dissection of putative mechanisms mediating constraints exerted by the genome architecture on somatic cancer evolution may provide an algorithm for understanding and predicting as well as modifying somatic cancer evolution in individual patients.
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Affiliation(s)
- Albert Rübben
- Independent Institute of Systems Sciences Aachen , 52064 Aachen , Germany ; Department of Dermatology, RWTH Aachen University , 52074 Aachen , Germany
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Kama A, Tannenbaum E. Effect of the SOS response on the mean fitness of unicellular populations: a quasispecies approach. PLoS One 2010; 5:e14113. [PMID: 21152423 PMCID: PMC2994707 DOI: 10.1371/journal.pone.0014113] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/17/2008] [Accepted: 10/26/2010] [Indexed: 11/23/2022] Open
Abstract
The goal of this paper is to develop a mathematical model that analyzes the selective advantage of the SOS response in unicellular organisms. To this end, this paper develops a quasispecies model that incorporates the SOS response. We consider a unicellular, asexually replicating population of organisms, whose genomes consist of a single, double-stranded DNA molecule, i.e. one chromosome. We assume that repair of post-replication mismatched base-pairs occurs with probability , and that the SOS response is triggered when the total number of mismatched base-pairs is at least . We further assume that the per-mismatch SOS elimination rate is characterized by a first-order rate constant . For a single fitness peak landscape where the master genome can sustain up to mismatches and remain viable, this model is analytically solvable in the limit of infinite sequence length. The results, which are confirmed by stochastic simulations, indicate that the SOS response does indeed confer a fitness advantage to a population, provided that it is only activated when DNA damage is so extensive that a cell will die if it does not attempt to repair its DNA.
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Affiliation(s)
- Amit Kama
- Department of Chemistry, Ben-Gurion University of the Negev, Be'er-Sheva, Israel
| | - Emmanuel Tannenbaum
- Department of Chemistry, Ben-Gurion University of the Negev, Be'er-Sheva, Israel
- * E-mail:
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Zeldovich KB, Chen P, Shakhnovich EI. Protein stability imposes limits on organism complexity and speed of molecular evolution. Proc Natl Acad Sci U S A 2007; 104:16152-7. [PMID: 17913881 PMCID: PMC2042177 DOI: 10.1073/pnas.0705366104] [Citation(s) in RCA: 187] [Impact Index Per Article: 11.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/11/2007] [Indexed: 01/18/2023] Open
Abstract
Classical population genetics a priori assigns fitness to alleles without considering molecular or functional properties of proteins that these alleles encode. Here we study population dynamics in a model where fitness can be inferred from physical properties of proteins under a physiological assumption that loss of stability of any protein encoded by an essential gene confers a lethal phenotype. Accumulation of mutations in organisms containing Gamma genes can then be represented as diffusion within the Gamma-dimensional hypercube with adsorbing boundaries determined, in each dimension, by loss of a protein's stability and, at higher stability, by lack of protein sequences. Solving the diffusion equation whose parameters are derived from the data on point mutations in proteins, we determine a universal distribution of protein stabilities, in agreement with existing data. The theory provides a fundamental relation between mutation rate, maximal genome size, and thermodynamic response of proteins to point mutations. It establishes a universal speed limit on rate of molecular evolution by predicting that populations go extinct (via lethal mutagenesis) when mutation rate exceeds approximately six mutations per essential part of genome per replication for mesophilic organisms and one to two mutations per genome per replication for thermophilic ones. Several RNA viruses function close to the evolutionary speed limit, whereas error correction mechanisms used by DNA viruses and nonmutant strains of bacteria featuring various genome lengths and mutation rates have brought these organisms universally approximately 1,000-fold below the natural speed limit.
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Affiliation(s)
| | - Peiqiu Chen
- Departments of Chemistry and Chemical Biology and
- Physics, Harvard University, 12 Oxford Street, Cambridge, MA 02138
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Lee B, Tannenbaum E. Asexual and sexual replication in sporulating organisms. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:021909. [PMID: 17930067 DOI: 10.1103/physreve.76.021909] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/30/2006] [Revised: 04/18/2007] [Indexed: 05/25/2023]
Abstract
Replication via sporulation is the replication strategy for all multicellular life, and may even be observed in unicellular life (such as with budding yeast). We consider diploid populations replicating via one of two possible sporulation mechanisms. (1) Asexual sporulation, whereby adult organisms produce single-celled diploid spores that grow into adults themselves. (2) Sexual sporulation, whereby adult organisms produce single-celled diploid spores that divide into haploid gametes. The haploid gametes enter a haploid "pool," where they may recombine with other haploids to form a diploid spore that then grows into an adult. We consider a haploid fusion rate given by second-order reaction kinetics. We work with a simplified model where the diploid genome consists of only two chromosomes, each of which may be rendered defective with a single point mutation of the wild-type. We find that the asexual strategy is favored when the rate of spore production is high compared to the characteristic growth rate from a spore to a reproducing adult. Conversely, the sexual strategy is favored when the rate of spore production is low compared to the characteristic growth rate from a spore to a reproducing adult. As the characteristic growth time increases, or as the population density increases, the critical ratio of spore production rate to organism growth rate at which the asexual strategy overtakes the sexual one is pushed to higher values. Therefore, the results of this model suggest that, for complex multicellular organisms, sexual replication is favored at high population densities and low growth and sporulation rates.
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Affiliation(s)
- Bohyun Lee
- School of Biology, Georgia Institute of Technology, Atlanta, Georgia 30332, USA
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Affiliation(s)
- Jesse Summers
- Department of Molecular Genetics and Microbiology, University of New Mexico, Albuquerque, NM 87131, USA.
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Tannenbaum E. Selective advantage for sexual reproduction. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:061925. [PMID: 16906882 DOI: 10.1103/physreve.73.061925] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/02/2005] [Revised: 12/08/2005] [Indexed: 05/11/2023]
Abstract
This paper develops a simplified model for sexual reproduction within the quasispecies formalism. The model assumes a diploid genome consisting of two chromosomes, where the fitness is determined by the number of chromosomes that are identical to a given master sequence. We also assume that there is a cost to sexual reproduction, given by a characteristic time tau(seek) during which haploid cells seek out a mate with which to recombine. If the mating strategy is such that only viable haploids can mate, then when tau(seek) = 0, it is possible to show that sexual reproduction will always out compete asexual reproduction. However, as tau(seek) increases, sexual reproduction only becomes advantageous at progressively higher mutation rates. Once the time cost for sex reaches a critical threshold, the selective advantage for sexual reproduction disappears entirely. The results of this paper suggest that sexual reproduction is not advantageous in small populations per se, but rather in populations with low replication rates. In this regime, the cost for sex is sufficiently low that the selective advantage obtained through recombination leads to the dominance of the strategy. In fact, at a given replication rate and for a fixed environment volume, sexual reproduction is selected for in high populations because of the reduced time spent finding a reproductive partner.
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Affiliation(s)
- Emmanuel Tannenbaum
- Department of Chemistry, Ben-Gurion University of the Negev, Be'er-Sheva 84105, Israel.
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Tannenbaum E, Sherley JL, Shakhnovich EI. Semiconservative quasispecies equations for polysomic genomes: the haploid case. J Theor Biol 2006; 241:791-805. [PMID: 16527313 DOI: 10.1016/j.jtbi.2006.01.016] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/10/2005] [Revised: 12/12/2005] [Accepted: 01/13/2006] [Indexed: 11/23/2022]
Abstract
This paper develops the semiconservative quasispecies equations for genomes consisting of an arbitrary number of chromosomes. We assume that the chromosomes are distinguishable, so that we are effectively considering haploid genomes. We derive the quasispecies equations under the assumption of arbitrary lesion repair efficiency, and consider the cases of both random and immortal strand chromosome segregation. We solve the model in the limit of infinite sequence length for the case of the static single fitness peak landscape, where the master genome has a first-order growth rate constant of k>1, and all other genomes have a first-order growth rate constant of 1. If we assume that each chromosome can tolerate an arbitrary number of lesions, so that only one master copy of the strands is necessary for a functional chromosome, then for random chromosome segregation we obtain an equilibrium mean fitness of [equation in text] below the error catastrophe, while for immortal strand co-segregation we obtain kappa (t=infinity)=k[e(-mu(1-lambda/2))+e(-mulambda/2)-1] (N denotes the number of chromosomes, lambda denotes the lesion repair efficiency, and mu is identical with epsilonL, where epsilon is the per base-pair mismatch probability, and L is the total genome length). It follows that immortal strand co-segregation leads to significantly better preservation of the master genome than random segregation when lesion repair is imperfect. Based on this result, we conjecture that certain classes of tumor cells exhibit immortal strand co-segregation.
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Affiliation(s)
- Emmanuel Tannenbaum
- Department of Chemistry, Ben-Gurion University of the Negev, Be'er-Sheva 84105, Israel.
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Brumer Y, Michor F, Shakhnovich EI. Genetic instability and the quasispecies model. J Theor Biol 2006; 241:216-22. [PMID: 16386760 DOI: 10.1016/j.jtbi.2005.11.018] [Citation(s) in RCA: 29] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/14/2005] [Revised: 11/09/2005] [Accepted: 11/15/2005] [Indexed: 11/23/2022]
Abstract
Genetic instability is a defining characteristic of cancers. Microsatellite instability (MIN) leads to by elevated point mutation rates, whereas chromosomal instability (CIN) refers to increased rates of losing or gaining whole chromosomes or parts of chromosomes during cell division. CIN and MIN are, in general, mutually exclusive. The quasispecies model is a very successful theoretical framework for the study of evolution at high mutation rates. It predicts the existence of an experimentally verified error catastrophe. This catastrophe occurs when the mutation rates exceed a threshold value, the error threshold, above which replicative infidelity is incompatible with cell survival. We analyse the semiconservative quasispecies model of both MIN and CIN tumors. We consider the role of post-methylation DNA repair in tumor cells and demonstrate that DNA repair is fundamental to the nature of the error catastrophe in both types of tumors. We find that CIN introduces a plateau in the maximum viable mutation rate for a repair-free model, which does not exist in the case of MIN. This provides a plausible explanation for the mutual exclusivity of CIN and MIN.
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Affiliation(s)
- Yisroel Brumer
- Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA 02138, USA.
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Tannenbaum E, Shakhnovich EI. Semiconservative replication, genetic repair, and many-gened genomes: Extending the quasispecies paradigm to living systems. Phys Life Rev 2005. [DOI: 10.1016/j.plrev.2005.08.001] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
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Tannenbaum E, Sherley JL, Shakhnovich EI. Evolutionary dynamics of adult stem cells: comparison of random and immortal-strand segregation mechanisms. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 71:041914. [PMID: 15903708 DOI: 10.1103/physreve.71.041914] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/28/2004] [Indexed: 05/02/2023]
Abstract
This paper develops a point-mutation model describing the evolutionary dynamics of a population of adult stem cells. Such a model may prove useful for quantitative studies of tissue aging and the emergence of cancer. We consider two modes of chromosome segregation: (1) random segregation, where the daughter chromosomes of a given parent chromosome segregate randomly into the stem cell and its differentiating sister cell and (2) "immortal DNA strand" co-segregation, for which the stem cell retains the daughter chromosomes with the oldest parent strands. Immortal strand co-segregation is a mechanism, originally proposed by [Cairns Nature (London) 255, 197 (1975)], by which stem cells preserve the integrity of their genomes. For random segregation, we develop an ordered strand pair formulation of the dynamics, analogous to the ordered strand pair formalism developed for quasispecies dynamics involving semiconservative replication with imperfect lesion repair (in this context, lesion repair is taken to mean repair of postreplication base-pair mismatches). Interestingly, a similar formulation is possible with immortal strand co-segregation, despite the fact that this segregation mechanism is age dependent. From our model we are able to mathematically show that, when lesion repair is imperfect, then immortal strand co-segregation leads to better preservation of the stem cell lineage than random chromosome segregation. Furthermore, our model allows us to estimate the optimal lesion repair efficiency for preserving an adult stem cell population for a given period of time. For human stem cells, we obtain that mispaired bases still present after replication and cell division should be left untouched, to avoid potentially fixing a mutation in both DNA strands.
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Affiliation(s)
- Emmanuel Tannenbaum
- Department of Chemistry and Chemical Biology, Harvard University, Cambridge, Massachusetts 02138, USA.
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