Simulation as a Tool to Identify Dynamical Typology of Water Frog Hemiсlonal Population Systems Dmytro Shabanov1, Marina Vladymyrova1, Anton Leonov2, Olga Biriuk1, Marina Kravchenko1, Quentin Mair3, Olena Meleshko4, Julian Newman3, Olena Usova1 and Grygoriy Zholtkevych1 1 V.N. Karazin Kharkiv National University, Ukraine 2 AppScale Systems Ink. 3 Glasgow Caledonian University, Scotland, UK 4 University Museum, Norwegian University of Science and Technology, Norway shabanov@karazin.ua Abstract. Some related species give rise to interspecies hybrids with hemiclonal inheritance. The gametes of such hybrids transfer the set of hereditary infor- mation of one of the parental species. The water frog, Pelophylax esculentus, is an example of such hybrids. The hemiclonal hybrids together with their parental species form a biosystem for which the suggested name is Hemiclonal Population System (HPS). The phenomenon of interspecific hemiclonal reproduction of wa- ter frogs has been intensively explored for several decades, but insufficient study has been devoted to the mechanisms of the composition constancy and ecological stability of their population systems. In this paper we focus on sustainability and possible results of transformations (in terms of different genetic forms and of population dynamics of each of these forms) of HPSs that consist of diploid representatives. By means of a simulation model we evaluated the long-term consequences of parameters that were derived from empirical research on a natural HPS at a particular brief time-period. This research was carried out in the region of eastern Ukraine, the so-called Siverskyi Donets center of diversity of the Pelophylax esculentus complex. Our study de- scribes the development of a dynamic typology of the HPS by principal compo- nent analysis of data generated by the simulations. We investigate the space of the possible states of the HPS. The simulations helped to split this state-space into six areas of stability, each of which corresponds to a different type of stabil- ity. Before conducting the simulation study we assumed there were only four truly stable states. Two new states were identified as a result of using this model. Keywords: Dynamic Typology, Hemiclonal Inheritance, Pelophylax esculentus complex, Simulation Modelling. 1 Introduction 1.1 Motivation The processes of transformation of natural biological systems (populations, ecosystems etc.) are an important challenge for modern science. In many cases these processes are not readily available for direct study (e.g. because of their long duration). In such cases we have to reconstruct their transformation models based on (1) the observed diversity of their natural states, (2) our hypothesis of what controls the change of such states, and (3) available empirical data. A simulation model plays a key role in such studies. Thanks to it, we can determine the possible directions of transformation of the target systems and build their typology. One of the insufficiently studied types of biological systems are those that create species capable of interspecies hybridization and their hemiclonal hybrids. Reproduction of hemiclonal hybrids occurs in biological systems which have been described as Hemiclonal Population Systems (HPSs) [15]. Water frogs are a typical model organism for studying such systems. This paper describes a computer simulation study of Hemiclonal Population Systems (HPS) of water frogs. First we introduce the peculiar characteristics of these biosys- tems. In the next section we indicate the advantages of computer simulation as a mod- elling tool together with the specific characteristics of the model presented here. Com- puter simulation has been described as the collaboration between experimenting and modelling [7]. The existing empirical data was used to determine model parameters and to assess the performance of the model against real-world situations. We describe ex- periments conducted upon the model itself and the way in which these can supplement fieldwork findings. 1.2 Reproduction within HPSs Most life forms which arise from sexual reproduction exhibit the typical biosystem hi- erarchy in which organisms exist as part of populations that form species. In these pop- ulations, genetically unique individuals produce sex cells (gametes), which bear a unique genome (holistic unitary complex of hereditary information) resulting from re- combination of two parental genomes (Fig.1 A). The consequence of this is the exist- ence of a population gene pool (common pool of genes). Populations of organisms with clonal reproduction, on the other hand, consist of clones, i.e. sets of genetically identical organisms (with accuracy limited by the error rate of copying during reproduction). These populations are presented as a set of relatively isolated lines consisting of genet- ically identical maternal and child individuals. This work is dedicated to a relatively rare type of population reproduction which differs from the two mentioned options. It is characteristic of hybridogenic complexes of species. This complex consists of two parental frog species [11]. Their full zoological names are Pelophylax lessonae (Camerano, 1882) and Pelophylax ridibundus (Pallas, 1771). They cross to produce hybrids named analogous to the species' name, e.g. Pelo- phylax esculentus (Linnaeus, 1758). Fig. 1. A comparison of the life cycle with fertilization and meiosis, which is characteristic of organisms with sexual reproduction (A), and modification of such a cycle, which is characteristic of interspecific hybrids with hemiclonal inheritance (B). The differences between the genomes of the parental species lead to the fact that pro- duction of normal gametes through recombination becomes impossible. Breeding hy- brids is made possible by specific changes in their life cycle (Fig. 1B). One parent ge- nome of such hybrids is eliminated from the germline cells (i.e. those cells which sub- sequently form the sex cells, gametes). Thereafter, the second parental genome under- goes endoreduplication (i.e. doubles without cell division). This gives rise to cells with two genomes that are identical (with accuracy limited by the errors of copying). They form genetically identical gametes. Hemiclonal hybrids differ both from usual organisms with recombinant reproduction and from clonal organisms. A HPS, where hemiclonal hybrids are reproduced, differs from normal recombinant and from clonal populations. The features of a HPS are as follows [15]: — Cooperative reproduction of individuals that differ in species composition of their genomes (i.e. representatives of parental species and hybrids of various genome compositions); — Vertical transmission of the clonal genome lines, which can combine with other recombinant or clonal genomes; — In the HPS, which includes individuals of the parental species, these individuals support the existence of a pool of recombining genes (gene pool), which corresponds to the gene pool of conventional monospecific populations; — Cases of limited interspecies recombination are observed (i.e. transferring of frag- ments of genetic information from the genome of one parental species into the gene pool of another parental species). The most common typology of a HPS is associated with an indication of the parental species and diploid hybrids and/or triploid hybrids, which are included in their compo- sition. The presence of P. esculentus is indicated by the letter E, letter L stands for P. lessonae, and letter R for P. ridibundus. The presence of polyploid P. esculentus is denoted by the letter p. Thus, L-E-HPS consists of P. lessonae and diploid P. esculen- tus, and R-E-HPS consists of P. ridibundus and diploid P. esculentus. Clonality of a genome is designated by putting its symbol in brackets. The water frogs' sex determi- nation system is similar to that in humans. The structure of the female genome includes the sex chromosome X. Individuals that have two female genomes are female (♀). The structure of the male genome includes the sex chromosome Y. Individuals with one female and one male genome are male (♂). Consider the simplest example of a L-E-HPS (Fig. 2), which includes a parent spe- cies (P. lessonae) as well as hybrids which clonally transmit the female P. ridibundus genome (XR). Hybrids are reproduced when crossed with the parental species individ- uals: ♀XL(XR)×♂XLYL → ♀XL(XR) : ♂YL(XR); ♀XLXL×♂YL(XR) → ♀XL(XR). Fig. 2. The occurrence of Pelophylax esculentus due to hybridization of Pelophylax lessonae with Pelophylax ridibundus and reproduction of P. esculentus with the parental species in L-E-HPS. Different regions of the P. esculentus complex distribution are characterized by HPSs of various compositions [2, 10, 11]. In some regions there are P. esculentus individuals which simultaneously produce gametes (L) and (R). This phenomenon is called hybrid amphispermy. Such individuals are referred as (L)(R). By crossing hybrids, which transmit genomes of the same parental species, there may occur representatives of these species. Typically, these individuals die before sex- ual maturity: ♀XL(XR)×♂YL(XR) → ♀XRXR → †. Different forms of frogs in a HPS differ in their vitality and fertility. The composition of zygotes, tadpoles and frogs of the different ages in the same HPS may vary signifi- cantly. Another feature of the hybridogenic complex of water frogs (not considered in this study) is that in some regions there are not only diploid hybrids (i.e., having two genomes), but hybrids with three (LLR or LRR) and even four (e.g. LLRR) genomes [2, 15]. Hemiclonal hybridization, the consequence of which is the occurrence of a HPS, is observed not only for the water frogs, but also for some other species groups [1]. HPSs of hybridogenic species complexes are a little-known category of biosystems. Their study, part of which is a simulation of their transformations, should lead to im- portant results. For example, the particular genome-selective elimination in interspe- cific hybrids may open new opportunities in biotechnology and genetic medicine. This allows, if necessary, the removal of unwanted fragments of the genome. Hemiclonal inheritance supports all offspring from crossing of two individuals (each of which trans- mits a clonal genome) to be genetically identical, i.e. clonal. The resulting organisms, which inherit clonal genomes of two different parent species, may be useful for bio- technology and agriculture. Study of the reason of the hemiclonal hybridization occur- rence and the effects of this phenomenon on the evolution of hybridizing species are of considerable theoretical interest. Although study of HPS transformations per se cannot solve these problems, it allows one to better understand mechanisms of appearance and maintenance of the stability of a biological system in which such amazing genetic phe- nomena are possible. For further study, it is necessary to describe the variety of possible states of HPSs, their dynamics, regularities and the conditions under which they are stable. Direct study of HPSs faces significant challenges. Determination of their composi- tion and reproduction mechanisms is associated with a significant amount of fieldwork. HPSs are relatively unusual and differ from well known biosystems such as biocenoses and sexually or clonally reproducing populations. Processes of change in HPS are quite extended temporally and may take decades. There is a high degree of variability in the composition of different HPSs and some are unique objects whose occurrence is highly unlikely. The study of only one unique HPS may not be enough for understanding the regularities of their dynamics. Additionally, HPSs are complex systems containing many stochastically interacting components. Some authors have used analytical modelling to describe the dynamics of a popula- tion system of water frogs [14, etc.]. Analytical models are suitable for the study of separate aspects of the HPS dynamics, however their usage faces significant challenges due to fact that a HPS is governed by an interrelated set of stochastic processes. We therefore consider simulation to be a more useful tool to study the general properties of HPSs. Christiansen [4] simulated reproduction of P. esculentus. Christiansen’s model is deterministic whereas our model allows modelling of random events in competition and breeding of animals. The works of Bove et al. [3] and Quilodran et al. [12] are recent studies based on the simulation of water frogs' HPSs. These studies consider the stabil- ity of specific types of HPSs under certain conditions. Our work, in contrast, seeks to analyze all possible stable states of a certain category of HPSs. Differences of our simulation from others are as follows. At the same time, we con- sider typical for most populations demographic factors (non-competitive and competi- tive mortality, differences in the reproduction probability, changes in viability with age) and the unique features of HPS. Our work considers the entire space of possible states of a certain category of HPS. Finally, we use population parameters, the estimates of which were obtained during field studies of this category of HPS. 1.3 Scope of Modelling Several authors of this work have previously built a deterministic discrete-time simu- lation model [5]. It enabled investigators to determine the population composition for a specified number of simulation steps, commencing with the current HPS composition. The objective of the current work is identification of the set of stable states of a HPS (consisting solely of diploid water frog individuals) using a stochastic simulation. The research was conducted in two stages. In the first stage a number of simulation experiments were performed for various initial compositions of the model HPS. The aim of this stage was to get insight into the possible end states for the HPSs. The second stage analyzed and classified these end states. The dynamic typology of the system was constructed based on the various initial compositions of the HPSs, the observed final states, and the observed transitions from initial to final states. Dynamic typology is based not only on analysis of the observed object states, but, above all, on a forecast of their future dynamics [9]. In this respect, dynamic typology differs both from associative typology (identification of groups of objects, related to one or more samples) and from analytical typology (partitioning of a set of objects into groups depending on the state of their observable characteristics) [15]. Hybrid frogs are reproduced differently in different regions [10, 14]. Before the problem of the dynamical typology is solved in general, for all possible types of HPSs and all known genetic forms of hybrids and their reproduction, it should be solved for one selected region. Authors of the current paper have previously described the Siv- erskyi Donets center of diversity of the Pelophylax esculentus complex located at the eastern Ukraine [15]. It is characterized by a high diversity of HPSs and provides the potential to test the adequacy of the model. The distribution of HPSs, including poly- ploid hybrids (not discussed in this paper), is related to the flood plain of the river Siv- erskyi Donets, and to its small tributaries and ponds located nearby within the Kharkiv and Donetsk regions. Within the Mzha and the Uda river basins (right tributaries of the Siverskyi Donets), HPSs consisting exclusively of diploids are widespread. The nature of gamete production in diploids from the Siverskyi Donets and Mzha and the Uda river basins is similar. We plan to describe the total variety of HPSs from the Siverskyi Donets center of diversity of the Pelophylax esculentus complex. As the first step, we consider a variety of HPSs located in the Mzha and the Uda river basins which consist of diploid repre- sentatives. 2 Model Description and its Justification 2.1 The Cycle of the Model A simulation model, as opposed to an analytical model, describes the process of the state transformations during time, not the dependence of the future states of the system on the current one. Therefore, for model development it is sufficient to describe the algorithm of changes in the HPS sizes with time. In this model, time is divided into discrete steps through which the model cycles. The cycle of the model corresponds to the calendar year. It sets the sequence of transformations: αng a → βng a → γng a → δng a → ωng a . Here αng a , βng a , γng a , δng a and ωng a is a sequence of transformations of the individual's groups of the genotype (g) and of the age (a) which correspond to the dif- ferent stages of the annual cycle (Table 1). Of k genotypes, x is female and y is male one. At a certain stage, the females of genotype f a and males of genotypes m a form a pair then descendants appear. The other symbols are explained in the description of the input parameters of the model. Table 1. A cycle of the model work. Changing the number of Symbol Meaning Transformation individuals in the groups α g ω g The initial numbers of Start of cycle t n a = t-1 n a α g individuals in groups of a na certain genotypes and age in Transition individuals β g α g t n a = t n a-1 each cycle to the next age The numbers of individuals in β g na groups with allowance for their transition to the next age Non-competitive γ β g tn a ≈ tn a × s a g g The numbers of individuals in death γ g na groups after non-competitive mortality δ g γ g g Immigration t n a = t n a+ t i a δ g The numbers of individuals in Algorithm of the na groups after joining immigrants Competitive reduction calculation ω t ng a based on in number of δ g g g t n a , c a , d a and V (it is The numbers of individuals in individuals described below) ω g groups after competitive na Algorithm of the mortality due to lack of resources calculation P(ff a ,mm a' ) Creation of the based on ωnf a , lf a , ωnm a' parental pairs and em a' (it is described Number of pairs of a certain separately below) P(ff a ,mm a' ) composition Reproduction ω g tn 0= Σ( P(ff a ,mm a' )× o (f ,m )×bf a ×wm a' ) g f m ω g tn 0 The number of offspring (a=0) End of cycle 2.2 Input Parameters Viability parameters. sga∈[0,1] — survival. This is proportion of individuals, which happens to be saved in a result of non-competitive death. If а>=maxaga, then sga=0 where max g a a stands for the maximum life span, and а is the age of individual. сg a ∈[0,1] — competitiveness factor. Let’s designate the probability of an individual to survive over the competitive reduction as с´g a . Denote maxс´ as the maximum value of this probability that is characteristic for representatives of the most competitive groups (0=0) — demand. This is the number of resources required for individual of a certain group. It’s magnitude is given for one cycle of the model. lf a ∈[0,1] — female loveliness. This is the female success rate in its search for a part- ner for reproduction. That is set in the same way as the competitiveness factor is spec- ified. Let’s denote the maximum value of the probability to find a partner as max l´, 0V and εD=V, then ω t ng a ≈ ε t ng a . If δD>V and εD>V, then ω t ng a ≈ ε t ng a ×V/ εD. If δD>V and εD 0; p♂XRYR + p♂XR(YL) + p♂YR(XL) + p♂(XL)(YR) + p♂(YL)(XR) > 0. Naturally, it does not make sense to consider the combinations in which the total proportion of genotypes involving into the HPS is not equal to unity, as well as those in which there are no male or female. The total number of genotypes that satisfy the above conditions equals 5895. Ten simulations of 500 steps were conducted for the 5895 starting points. The collection of all observed outcomes of the simulation was divided into types, depending on what kind of genotypes presented in the model HPS over 500 steps. For 4778 initial states, all 10 iterations led to any one outcome. In 1117 cases, outcomes were variable. For these 1117 initial states, 10 more runs were carried out; thus, the total number of simulations was 70,120. To determine the states, which the model HPS can move to, we examined the inter- vals between step 100 and 200, step 200 and 300, 300 and 400 as well as 400 and 500. The total number of intervals were 162,580; its number is less than possible one due to the model HPS, which dies at any stage of the simulation. To divide the obtained set of 70,120 final states of individual simulations into groups, we used the analysis of this aggregation by the method of principal components. The first and the second major components make it possible to divide the HPS into 6 groups (Fig. 3. A): — Extinction — 37,946 runs ended in HPS extinction; — R-E-HPS — 22,204 runs ended in different HPSs R-E-types; — E-HPS-type I — 5884 results; such HPS include ♀(XL)(XR) and ♂(XL)(YR); — E-HPS-type II — 3892 results corresponding to another possible type of the E-HPS, which include ♀(XL)(XR) and ♂(YL)(XR); — Extincting — 124 results located nearby of the point Extinction. They continue to end up in extinction. The structure of these finals can be further divided into 13 types; — R-population — 70 results, consisting only of the population P. ridibundus. The R-E-HPS group, which corresponds to 22,204 results by its composition, can be divided into three parts. Their relative positions can be seen in the plane of the first and the third main component (Fig. 3. B): — Stable R-E-HPS-type I — 18,852; — Stable R-E-HPS-type II — 2,847 results; — Indifferent R-E-HPS — 505 results in which P. ridibundus is present as well as P. esculentus, which transmits both genome P. lessonae and female genome P. ridi- bundus. Fig. 3. Ordination of the results of 70,120 simulations. A. Ordination of the results on the plane of the first two principal components. B. Ordination of the results on the plane of the first and third principal components. To determine the types of stability for the dynamic types of the allocated HPS, the 162 580 pairs of initial state and its outcome were analyzed. The observed stated were clas- sified by the types of stability as shown in Fig. 4. The observed states are divided into three groups (Equilibrium states, Transient states and Attractive states) as shown in Fig.6, though this division is rather subjective. The Indifferent equilibrium (II) can be considered as a transient state as well. Two ver- sions of the transient states are associated with the directed transitions to other states. Extincting state (IV) is associated with the transition to the extinction state (VI), and transforming stage (III) with the transition to other states. Fig. 4. Types of biosystems stability, observed in experiments with the simulation model. Six basins of sustainability exist in the space of possible states of HPS that consists of diploid individuals, and this corresponds to specific features of the Siverskyi Donets center of diversity of the Pelophylax esculentus complex by their character of the ge- nome transmission. One of them complies with the population of parental species. Two are E-HPS, i.e. spawning population of HPS consisting exclusively of P. esculentus. When crossing P. esculentus with the hybrid amphispermy, the offspring of parental species appearing in such systems dies before the age of maturity. E-HPS-type I: ♀(XL)(XR)×♂(XL)(YR)→XLXL:(XL)(XR):(XL)(YR):XRYR→♀(XL)(XR) : ♂(XL)(YR); E-HPS-type II: ♀(XL)(XR)×♂(YL)(XR)→XLYL:(XL)(XR):(YL)(XR):XRXR→♀(XL)(XR) : ♂(YL)(XR). There are two more basins of stability corresponding to the R-E-HPS containing P. ridibundus and P. esculentus, and the last basin is associated with extinction of the HPS. 5 Outcome Interpretation When assessing the results, one must be aware that the result of the simulation is not proof of a hypothesis. At the same time, using simulation models as an exploratory tool has a significant advantage over unformulated conceptual models. We do not have the sufficient empirical data to describe precisely the processes occurring in the natural HPS. The lack of empirical data is offset by a set of presumptions [13] and hypotheses. Simulation allows us to derive consequences arising from the set of initial assumptions. These sets of consequences may or may not contradict the observable empirical pic- ture. An observed contradiction is the basis for rejecting a set of initial assumptions or adjustments. Agreement is not an evidence for the initial assumptions, but can be seen as an argument in their favor, in other words as corroboration, not as proof (Fig. 5). Fig. 5. Using a simulation model to test hypotheses about the mechanisms of HPS functioning. The distribution of the outcomes of the model HPS agrees with the empirical data, and it is ob- tained in the case when variant B of the variable part of the model is selected. It does not prove variant B to be true, but it allows to discard variant A in its favor. Fig 5 shows how the simulation results can be used to select between different versions of the assumptions in the absence of empirical data. Model predictions regarding the expected diversity of HPS conditions, depends on assumptions, on which the variant part of the model was constructed. The predicted (modeled) expected diversity of HPS states can be compared to empirically observed diversity of HPS states. The results of this comparison can be seen as the arguments in favor of such initial assumptions that yielded close to the empirical distribution of states HPS. Simulation results, among other things, stimulate the collection of the empirical data. Comparison of predictions, obtained by modelling, and the available fragmented data about the composition of the natural HPS and character of P. esculentus gametogenesis shows that currently there is an absence of E-type HPS in the studied region, although that there are grounds for thinking that such a system probably existed in 1995 [6]. Studying the composition of tadpoles in this pond shows that there existed E-HPS-type II [8]. In fact, the majority of observed systems are in a transient state. The authors believe that this research is only a starting step in design of the dynamic typology of the water frogs’ HPS. The usefulness of the simulation is not limited to fact that two stable states, previously not known to the authors, can theoretically be found: R-E-HPS-type II and E-HPS-type II. Results of simulations allow developing a pro- gram for further research. The simulation results define the data collection program for testing the adequacy of the results. The currently available data do not contradict the model findings. Future work will include the study of unstable states of HPS that are observed in the natural environment [16], as well as the extension of the model by in- corporating hybrids with three genomes, which are specific for the Siverskyi Donets center of diversity of the Pelophylax esculentus complex. 6 Discussion and Conclusion We study the unusual category of biosystems, Hemiclonal Population Systems (HPS), through the example of the hybridogenic complex of water frogs, the Pelophylax escu- lentus complex. The unusual method of reproduction of interspecific hybrids within HPS results in their unusual features, which need more investigation. An important method for studying such systems is computer simulation. A simulation model of the water frogs’ HPS has been presented. The model inputs are the parameters describing the comparative vitality of various genetic forms of frogs, the results of their probable crossings, as well as the experimental conditions such as the capacity of the environment, the initial composition of the model HPS, and a sce- nario for the introduction of migrated frogs into the model HPS. Evaluation of the com- parative vitality and the crossing results are defined in accordance with the results of population-environmental research in the Siverskyi Donets center of diversity of the Pelophylax esculentus complex. Since direct empirical data was lacking, the values of the parameters used have been estimations that authors put forward on the basis of a study of this region. Repeated runs of the model yield a probability distribution of outcomes of various HPS transformations according to their initial states and experimenter-determined pa- rameters. 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