**2. Haploid Evolutionary Constructor methodology description**

The HEC methodology provides modeling and simulation of biological and evolutionary processes in trophically linked communities of unicellular haploid organisms. Figure 1 shows the scheme of the main HEC objects and processes. The HEC simulates trophic interconnected haploid organism networks that are combined into populations according to genetic proximity and reside in one whole volume termed *environment*. Organisms may consume and *utilize* substrates and synthesize, and then *secrete* products into the environment, which inversely can be used by other organisms as substrates. Some substrates favorably affect population growth, others, on the contrary, may have an inhibiting effect. The efficiencies (as rate constants) of substrate utilization and product production are controlled by certain genes.

P1

PM

SL-1

hand corner of the figure. The bounded area represents the environment.

**2.** produce the same variety of specific products;

substrate production rate, the third group (*ri*

**3.** have the same trophic strategy;

**4.** have the same synthesis strategy.

**1.** utilize the same variety of nonspecific and specific substrates;

We define basic terms and notions, used hereinafter as follows:

**2.2. Population modeling**

of inheritance;

SL

P2

S4

S 4

S3

S2

S1

Evolution, Biodiversity and Ecology in Microbial Communities: Mathematical Modeling and Simulation with…

P4

S 5

**Figure 1.** Scheme of HEC objects and processes. Blue circles represent populations P1, P2, … PM. Groups of colored dots represent substrates: specific S1, S2, … SL and nonspecific N1. Arrows between populations represent trophic rela‐ tionships: the population, from which the arrow comes, produces the substrate, used by the population to which this arrow leads. Red arrows represent a substrate's activating effect, blue arrows – an inhibiting effect. The usage of a nonspecific substrate is indicated by orange arrows. The flow action is represented by the thick arrow in the lower left-

We consider populations to be a set of cells, that have common substrate using and producing properties. Cells are considered to belong to the same population (same species) if they:

**• trait** – any given substrate synthesis or utilization rate constant. Every trait is considered as unambiguously controlled by one **gene**. In this particular case, the **gene** is considered a unit

**• individual's genotype** is the set of alleles, divided into five groups. The first group (*ci*

**• allele** – a gene variant, i.e. a particular value of the corresponding constant;

characterizes the efficiency of specific substrate utilization (*si*

N1

NR

P3

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)

) –

), the second group (*di*

) – efficiency of nonspecific substrate utilization,

So-called *nonspecific* substrates are reproduced by the in-flow (Figure 1 shows scheme with one nonspecific substrate N1). By contrast substrates running into the environment only through cell activity (synthesis and secretion) are known as *specific* (represented by Si in Figure 1).

#### **2.1. Environment**

Environment is a bounded flow system of fixed volume *Vtotal*, containing all populations and substrates. The environment is also a mediator of relationships between populations (and substrates); the inflowing and outflowing processes of both substrates and cells are connected with it. The environment is characterized by following variables:


and the following processes:

**substrates inflow into the environment** – increases a nonspecific substrate concentration according to the flow rate and concentration of these substrates in the flow;

**substrates outflow from the environment** – reduces both nonspecific and specific substrate concentrations according to the flow rate;

the inflow/outflow of nonspecific substrates follows the formulas:

$$N\_{env,i}(t+1) = N\_{env,i}(t) + k\_{env,flow} \cdot \left(N\_{flow,i} - N\_{env,i}(t)\right) \tag{1}$$

the specific substrates outflow follows the formula below (specific substrates flow into the environment associated with these substrates through cell synthesis as described below):

$$\mathcal{S}\_{env,i}(t+1) = \mathcal{S}\_{env,i}(t) \cdot \left(1 - k\_{env,flow}\right) \tag{2}$$

Evolution, Biodiversity and Ecology in Microbial Communities: Mathematical Modeling and Simulation with… http://dx.doi.org/10.5772/58302 115

**Figure 1.** Scheme of HEC objects and processes. Blue circles represent populations P1, P2, … PM. Groups of colored dots represent substrates: specific S1, S2, … SL and nonspecific N1. Arrows between populations represent trophic rela‐ tionships: the population, from which the arrow comes, produces the substrate, used by the population to which this arrow leads. Red arrows represent a substrate's activating effect, blue arrows – an inhibiting effect. The usage of a nonspecific substrate is indicated by orange arrows. The flow action is represented by the thick arrow in the lower lefthand corner of the figure. The bounded area represents the environment.

#### **2.2. Population modeling**

the scheme of the main HEC objects and processes. The HEC simulates trophic interconnected haploid organism networks that are combined into populations according to genetic proximity and reside in one whole volume termed *environment*. Organisms may consume and *utilize* substrates and synthesize, and then *secrete* products into the environment, which inversely can be used by other organisms as substrates. Some substrates favorably affect population growth, others, on the contrary, may have an inhibiting effect. The efficiencies (as rate constants) of

So-called *nonspecific* substrates are reproduced by the in-flow (Figure 1 shows scheme with one nonspecific substrate N1). By contrast substrates running into the environment only through cell activity (synthesis and secretion) are known as *specific* (represented by Si in

Environment is a bounded flow system of fixed volume *Vtotal*, containing all populations and substrates. The environment is also a mediator of relationships between populations (and substrates); the inflowing and outflowing processes of both substrates and cells are connected

**substrates inflow into the environment** – increases a nonspecific substrate concentration

**substrates outflow from the environment** – reduces both nonspecific and specific substrate

the specific substrates outflow follows the formula below (specific substrates flow into the environment associated with these substrates through cell synthesis as described below):

*N t N tk N N t env i*, , , ,, ( + 1) *env i env flow flow i env i* ) + ×( -= ( )( ) (1)

*Senv i*,,, ( 1) ( ) 1 = *Senv i t* ×( -+ *kt env flow* ) (2)

**• Nenv,i** – concentration of the ith nonspecific substrate in the environment (in mM);

**• Senv,i** – concentration of the ith specific substrate in the environment (in mM);

**• Nflow,i** – concentration of the ith nonspecific substrate in the flow (in mM);

according to the flow rate and concentration of these substrates in the flow;

the inflow/outflow of nonspecific substrates follows the formulas:

substrate utilization and product production are controlled by certain genes.

with it. The environment is characterized by following variables:

**• Vtotal** – environment capacity (in liters);

concentrations according to the flow rate;

and the following processes:

**• kkenv,flow** – flow rate (in % *Vtotal* per unit time);

Figure 1).

**2.1. Environment**

114 Biodiversity - The Dynamic Balance of the Planet

We consider populations to be a set of cells, that have common substrate using and producing properties. Cells are considered to belong to the same population (same species) if they:


We define basic terms and notions, used hereinafter as follows:


the fourth group (*mi* ) – efficiency of immunity against phages, and the fifth group (*vi* ) – phage (viruses) genes;

*N*

gene);

γ, γ0, γ<sup>i</sup>

growth;

spondent genes);

*P* – population size;

*kdeath* – population mortality rate;

(depends on corresponding traits).

of a new allele is possible, Figure 3).

*abasal* – "natural increase" of the population;

*S*

(in proportion to the population size) values;

*kflow* – flow rate in the environment ("washout" rate);

<sup>→</sup> – vector of specific substrates, consumed by the cells of the population from the environment

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*r0* – utilization rate for the unique nonspecific substrate (trait, controlled by the corresponding

→ – vector of corresponding specific substrate utilization rates (traits, controlled by corre‐

– coefficients, describing the nonlinear nature of substrate influence on population

*Kij* – coefficients, describing the efficiency of substrate influence on population growth

The equation [eq.3] governs the utilization process of several specific and one nonspecific substrate, where the latter is essential for cells – when it is not available, the population does not grow. Substrates have strong cooperativity. Besides, some substrates are able to compen‐ sate in some degree the lack of other necessary substrates, including a nonspecific substrate deficiency. The trophic strategy described in [eq.3], satisfies the **Rubel's law of replaceability**

The equation [eq.4] again governs the utilization process of several specific and one nonspecific substrate. Nevertheless, every substrate is essential. A deficiency of one substrate cannot be compensated by an excess others. The trophic strategy described by [eq.4] satisfies the ecological **Liebig's law of the minimum** [4] and called the **noncompensatory trophic strategy**. The equation [eq.5] governs the utilization process of one nonspecific substrate coupled with the inhibiting effect of specific substrates on population growth. Besides, this effect is cooper‐

Similar to the concept of a monomorphic population, we define the concept of **polymorphic population**, which can be regarded as a set of monomorphic subpopulations. Cells in a polymorphic population have the same gene variety, while different cells can have different alleles of one or several genes. The polymorphic population is characterized by the "general‐ ized genome" – a set of population's **genetic spectra**. The **genetic spectrum** is the distribution

Mutation in terms of a genetic spectrum means the change of its profile (thus, the formation

The change of the polymorphic population size is calculated according to the following scheme. The polymorphic population is split into many monomorphic subpopulations. Then,

**of ecological factors** [39], and is called the **compensatory trophic strategy**.

ative. This trophic strategy is called the **inhibitory trophic strategy**.

of allele occurrence frequencies in a population (for one gene) (Figure 2).

**• mutation** is a change of a corresponding trait value, which can be interpreted as a gene shifting into another state (allele).

Using the introduced terminology, the concept of a **monomorphic population** can be formed – a population of "genetically identical" cells, where all cells have corresponding genes represented by the same alleles. The genotype, common for all the cells of such population, is called the **monomorphic population genotype**.

In order for cells to use substrates for their reproduction and population size growth along with products synthesis, at first they have to get these substrates from environment. In the HEC, the process of substrate consumption is described by the particular step, where a cell's requirement of various substrates and the availability of substrates in the environment are taken into consideration. In case of any substrate deficiency, a competition for this substrate may occur either intrapopulation or interpopulation (when cells of several populations may use the same substrate at the same time). In case of substrate excess, the maximum amount of the substrate consumed by one cell is defined by the value of the substrate **consumption rate**. This value is species-specific and cannot be changed due to mutation (in this HEC version) – this may be equivalent to, for example, the size limit for one cell. Hence, monomorphic population is additionally characterized by the amount of substrate molecules consumed.

**Trophic strategies** is a term for the formulas and laws, describing population changes in a single generation depending on population size, the amount of consumed substrates, the flow rate, mortality rate and other factors. Examples of trophic strategies are illustrated by the equations below (other formulas may be also used, including those defined by the user):

$$
\Delta P = F\_1(\overrightarrow{\text{N}}, \overrightarrow{\text{S}}, \overrightarrow{\text{R}}, \overrightarrow{\text{C}}, \overrightarrow{\text{C}}, P) = \sqrt{r\_0 n\_0(P) \bullet \sum\_{i \in I\_{camad}} c\_i s\_i(P)} \cdot k\_{flow} \bullet P \cdot k\_{death} \bullet P^2 \tag{3}
$$

∆*P* = *F*<sup>2</sup> (*N* <sup>→</sup>, *<sup>S</sup>* <sup>→</sup> , *R* <sup>→</sup> , *<sup>C</sup>* <sup>→</sup> , *<sup>P</sup>*)=*<sup>P</sup>* <sup>∙</sup> ( *n* 0 *P K*01(*<sup>r</sup>* 0) ) *γ*0 <sup>1</sup> <sup>+</sup> ( *<sup>n</sup>* 0 *P K*02(*<sup>r</sup>* 0) ) *<sup>γ</sup>*<sup>0</sup> <sup>∙</sup> <sup>∏</sup> *<sup>i</sup>*∈*Iconsumed* ( *s i P Ki* 1(*<sup>c</sup> i* ) ) *γi* 1 + ( *s i P Ki* 2(*<sup>c</sup> i* ) ) *<sup>γ</sup><sup>i</sup>* - *<sup>k</sup> flow* <sup>∙</sup>*<sup>P</sup>* - *kdeath* <sup>∙</sup>*<sup>P</sup>* <sup>2</sup> (4)

$$
\Delta P = F\_3(\vec{\mathcal{N}}, \vec{\mathcal{S}}, \vec{\mathcal{R}}, \vec{\mathcal{C}}, P) = a\_{\text{basal}}(n\_0) \bullet P \cdot \sqrt{\sum\_{i \in I\_{avuud}} c\_i s\_i(P)} \cdot k\_{deadh} \bullet P^2 \tag{5}
$$

where

*Iconsumed* – set of indices of substrates consumed;

*n0* – amount of nonspecific substrate, consumed by the cells of population from the environ‐ ment (in proportion to the population size);

*N* <sup>→</sup> – vector of specific substrates, consumed by the cells of the population from the environment (in proportion to the population size) values;

*r0* – utilization rate for the unique nonspecific substrate (trait, controlled by the corresponding gene);

*S* → – vector of corresponding specific substrate utilization rates (traits, controlled by corre‐ spondent genes);

*P* – population size;

the fourth group (*mi*

116 Biodiversity - The Dynamic Balance of the Planet

∆*P* = *F*<sup>1</sup>

(*N* <sup>→</sup>, *<sup>S</sup>* <sup>→</sup> , *R* <sup>→</sup> , *<sup>C</sup>*

∆*P* = *F*<sup>3</sup>

(*N* <sup>→</sup>, *<sup>S</sup>* <sup>→</sup> , *R* <sup>→</sup> , *<sup>C</sup>*

*Iconsumed* – set of indices of substrates consumed;

ment (in proportion to the population size);

∆*P* = *F*<sup>2</sup>

where

(*N* <sup>→</sup>, *<sup>S</sup>* <sup>→</sup> , *R* <sup>→</sup> , *<sup>C</sup>*

<sup>→</sup> , *<sup>P</sup>*)= *<sup>r</sup>*0*n*<sup>0</sup>

( *n* 0 *P K*01(*<sup>r</sup>* 0) ) *γ*0

<sup>1</sup> <sup>+</sup> ( *<sup>n</sup>* 0 *P K*02(*<sup>r</sup>* 0) )

<sup>→</sup> , *<sup>P</sup>*)=*<sup>P</sup>* <sup>∙</sup>

shifting into another state (allele).

called the **monomorphic population genotype**.

(viruses) genes;

) – efficiency of immunity against phages, and the fifth group (*vi*

**• mutation** is a change of a corresponding trait value, which can be interpreted as a gene

Using the introduced terminology, the concept of a **monomorphic population** can be formed – a population of "genetically identical" cells, where all cells have corresponding genes represented by the same alleles. The genotype, common for all the cells of such population, is

In order for cells to use substrates for their reproduction and population size growth along with products synthesis, at first they have to get these substrates from environment. In the HEC, the process of substrate consumption is described by the particular step, where a cell's requirement of various substrates and the availability of substrates in the environment are taken into consideration. In case of any substrate deficiency, a competition for this substrate may occur either intrapopulation or interpopulation (when cells of several populations may use the same substrate at the same time). In case of substrate excess, the maximum amount of the substrate consumed by one cell is defined by the value of the substrate **consumption rate**. This value is species-specific and cannot be changed due to mutation (in this HEC version) – this may be equivalent to, for example, the size limit for one cell. Hence, monomorphic population is additionally characterized by the amount of substrate molecules consumed.

**Trophic strategies** is a term for the formulas and laws, describing population changes in a single generation depending on population size, the amount of consumed substrates, the flow rate, mortality rate and other factors. Examples of trophic strategies are illustrated by the equations below (other formulas may be also used, including those defined by the user):

> (*P*)∙ ∑ *i*∈*Iconsumed ci si*

> > *<sup>γ</sup>*<sup>0</sup> <sup>∙</sup> <sup>∏</sup> *<sup>i</sup>*∈*Iconsumed*

*n0* – amount of nonspecific substrate, consumed by the cells of population from the environ‐

<sup>→</sup> , *<sup>P</sup>*)=*abasal*(*n*0)∙*<sup>P</sup>* - <sup>∑</sup> *<sup>i</sup>*∈*Iconsumed*

( *s i P Ki* 1(*<sup>c</sup> i* ) ) *γi*

1 + ( *s i P Ki* 2(*<sup>c</sup> i* ) )

> *ci si*

(*P*) - *<sup>k</sup> flow* <sup>∙</sup>*<sup>P</sup>* - *kdeath* <sup>∙</sup>*<sup>P</sup>* <sup>2</sup> (3)

*<sup>γ</sup><sup>i</sup>* - *<sup>k</sup> flow* <sup>∙</sup>*<sup>P</sup>* - *kdeath* <sup>∙</sup>*<sup>P</sup>* <sup>2</sup> (4)

(*P*) - *kdeath* <sup>∙</sup>*<sup>P</sup>* <sup>2</sup> (5)

) – phage

*kflow* – flow rate in the environment ("washout" rate);

*kdeath* – population mortality rate;

*abasal* – "natural increase" of the population;

γ, γ0, γ<sup>i</sup> – coefficients, describing the nonlinear nature of substrate influence on population growth;

*Kij* – coefficients, describing the efficiency of substrate influence on population growth (depends on corresponding traits).

The equation [eq.3] governs the utilization process of several specific and one nonspecific substrate, where the latter is essential for cells – when it is not available, the population does not grow. Substrates have strong cooperativity. Besides, some substrates are able to compen‐ sate in some degree the lack of other necessary substrates, including a nonspecific substrate deficiency. The trophic strategy described in [eq.3], satisfies the **Rubel's law of replaceability of ecological factors** [39], and is called the **compensatory trophic strategy**.

The equation [eq.4] again governs the utilization process of several specific and one nonspecific substrate. Nevertheless, every substrate is essential. A deficiency of one substrate cannot be compensated by an excess others. The trophic strategy described by [eq.4] satisfies the ecological **Liebig's law of the minimum** [4] and called the **noncompensatory trophic strategy**.

The equation [eq.5] governs the utilization process of one nonspecific substrate coupled with the inhibiting effect of specific substrates on population growth. Besides, this effect is cooper‐ ative. This trophic strategy is called the **inhibitory trophic strategy**.

Similar to the concept of a monomorphic population, we define the concept of **polymorphic population**, which can be regarded as a set of monomorphic subpopulations. Cells in a polymorphic population have the same gene variety, while different cells can have different alleles of one or several genes. The polymorphic population is characterized by the "general‐ ized genome" – a set of population's **genetic spectra**. The **genetic spectrum** is the distribution of allele occurrence frequencies in a population (for one gene) (Figure 2).

Mutation in terms of a genetic spectrum means the change of its profile (thus, the formation of a new allele is possible, Figure 3).

The change of the polymorphic population size is calculated according to the following scheme. The polymorphic population is split into many monomorphic subpopulations. Then,

the other populations. For this reason, we consider two forms (states) of substrates in the modeling of the internal cell substrates. The first form are the substrates that are "**ready for utilization**", the second form are **synthesized** substrates. The substrates of the first group are replenished by substrate consumption from the environment and through transition of synthesized substrates (if the cell can synthesize it). During reproduction and product synthesis, the substrates of the first group are expended. The substrates of the second group are replenished only by synthesis. A principal scheme of cell metabolism in a population, consisting of the stages of substrate consumption, utilization, synthesis, and secretion is

Evolution, Biodiversity and Ecology in Microbial Communities: Mathematical Modeling and Simulation with…

**Figure 4.** A scheme of trophic processes through the example of one cell. The cell utilizes three types of substrates and synthesises two types of substrates, one of which it can utilize itself. A synthesized substrate, which cannot be utilized by the cell (yellow) is comprehensively secreted into the environment, while the second substrate (red) is partly used

Product synthesis by the cells of a polymorphic population is described by the gene network

model of metabolite synthesis, which we call the **synthesis strategy** (Figure 5).

∆*si* = *P* ∙ ∑

;

*j*∈*Spectr*<sup>i</sup> *dij si*(*<sup>P</sup> <sup>j</sup>*

**Utilization genes**

**Substrate consumption**

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119

**Product** 

*<sup>P</sup>*) (6)

**secretion Synthesis genes**

illustrated by (Figure 4).

**Substrate consumption**

where

*Δs<sup>i</sup>*

**Product secretion**

by the cell itself (consequently, there is less or no secretion into the environment).

An example of a formula describing a simple synthesis strategy:

– amount of synthesized i-type substrate;

*dij* – trait value in genetic spectra *Spectri*

**Figure 2.** The genetic spectrum shows that the given trait value is 2 for 20% of individuals in this population, 3 for 50%, and 4 for 30%.

**Figure 3.** Change of genetic spectrum due to mutations.

the growth for each subpopulation is calculated according to formulas similar to the above‐ mentioned (eq.3-5). Next, the monomorphic populations are merged into a polymorphic population. The polymorphic population is split based on the proportion of every allele for all genes in the population genotype. The substrates consumed by the polymorphic population are divided proportional to the sizes of the single monomorphic populations. It noteworthy, that each single monomorphic population growth may differ markedly from the growth of other subpopulations. It depends on the monomorphic population genotype, population size and the amount of certain substrates consumed by the population. Consequently, the propor‐ tion of alleles in a population may change (which may be interpreted as the adaptation of the population to certain conditions).

#### **2.3. Metabolism**

When a cell synthesizes a product, that it can utilize itself, it is obvious that there is "no use" in secreting this product into the environment and then competing for it on "equal terms" with the other populations. For this reason, we consider two forms (states) of substrates in the modeling of the internal cell substrates. The first form are the substrates that are "**ready for utilization**", the second form are **synthesized** substrates. The substrates of the first group are replenished by substrate consumption from the environment and through transition of synthesized substrates (if the cell can synthesize it). During reproduction and product synthesis, the substrates of the first group are expended. The substrates of the second group are replenished only by synthesis. A principal scheme of cell metabolism in a population, consisting of the stages of substrate consumption, utilization, synthesis, and secretion is illustrated by (Figure 4).

**Figure 4.** A scheme of trophic processes through the example of one cell. The cell utilizes three types of substrates and synthesises two types of substrates, one of which it can utilize itself. A synthesized substrate, which cannot be utilized by the cell (yellow) is comprehensively secreted into the environment, while the second substrate (red) is partly used by the cell itself (consequently, there is less or no secretion into the environment).

Product synthesis by the cells of a polymorphic population is described by the gene network model of metabolite synthesis, which we call the **synthesis strategy** (Figure 5).

An example of a formula describing a simple synthesis strategy:

$$
\Delta s\_i = P \bullet \sum\_{j \in \text{Spec}\tau\_i} d\_{ij} s\_i^{\{P\_j\}} \int\_P
$$

where

the growth for each subpopulation is calculated according to formulas similar to the above‐ mentioned (eq.3-5). Next, the monomorphic populations are merged into a polymorphic population. The polymorphic population is split based on the proportion of every allele for all genes in the population genotype. The substrates consumed by the polymorphic population are divided proportional to the sizes of the single monomorphic populations. It noteworthy, that each single monomorphic population growth may differ markedly from the growth of other subpopulations. It depends on the monomorphic population genotype, population size and the amount of certain substrates consumed by the population. Consequently, the propor‐ tion of alleles in a population may change (which may be interpreted as the adaptation of the

1 2 3 4 Alleles

**Figure 2.** The genetic spectrum shows that the given trait value is 2 for 20% of individuals in this population, 3 for

%

50

118 Biodiversity - The Dynamic Balance of the Planet

40

30

20

10

0

When a cell synthesizes a product, that it can utilize itself, it is obvious that there is "no use" in secreting this product into the environment and then competing for it on "equal terms" with

population to certain conditions).

**Figure 3.** Change of genetic spectrum due to mutations.

**2.3. Metabolism**

50%, and 4 for 30%.

*Δs<sup>i</sup>* – amount of synthesized i-type substrate;

*dij* – trait value in genetic spectra *Spectri* ;

#### *P* – population size;

*Pj* – proportion of individuals having *dij* trait value in population (in the genetic spectra).

Phages income into environment

For each population Pinfected : calculate a fraction of cells realizing lytic/lysogenic strategy as a function of cells' "fullness" by substrates

For each population: If population can be infected (either no immunity genes, or no phage genes incorporated)

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Generation of infected population : copy polymorphic genome, addition of phage genes \_ Infected population size Pinfected is proportional to Ppop×Pphage

Infestations of one cell requires a certain number of phages from environment (defined by parameter)

The choice of the lysogenic or the lytic scenario depends on the conditions of the cell population at the moment of infection: optimum conditions lead to a lytic type, pessimal conditions– to a lysogenic. In the latter case, a part of the population randomly switches to the lytic form if

Evolution, Biodiversity and Ecology in Microbial Communities: Mathematical Modeling and Simulation with…

The polymorphism of phages, the formation of new strains owing to mutations and competi‐ tion between strains are also described via the genetic spectra arithmetics. However, in the phage populations unlike prokaryotic populations, genes define a specific virulence (an ability

**3. Simulation of prokaryotic communities via the Haploid Evolutionary**

Through the use of the HEC software tool, we have simulated a number of biological models of the functioning and evolution of unicellular haploid organism communities [37]. Inter alia, the correspondence of the modeling results to biological data as well as previously published mathematical models has been illustrated. We estimated the key parameters of the model, regarding cell size, number of substrates required for cell division and other factors based on

conditions improve, causing the death of this part and phage generation (Figure 7).

to infect certain populations) and abundance (the number of copies per lysed cell).

•After lysis of infected cells

**Figure 6.** Scheme of phage infestation.

**Constructor**

*E.coli* cell information [40].

• From outside

**Figure 5.** Common pattern of synthesis strategies calculation.

#### **2.4. Infection process modeling**

To model a phage infection, the HEC provides extended objects of polymorphic populations: *polymorphic phage populations* and the *polymorphic population of infected cells* (the "normal" polymorphic populations are therefore regarded as "healthy").

The infection modeling includes the following phases: infection of the healthy cells through phage penetration from the environment into one part of the cell population, phage reproduction inside of the infected cells, and finally, the phages burst into the environ‐ ment after the lysis of cells. The infected cells form polymorphic populations, further reproduction of which may follow a lytic or a lysogenic pathway. The lytic pathway means death (lysis) of the infected cells with synchronous phage formation and their transporta‐ tion into the environment (the number of phages depends on their profusion). The lysogenic pathway means prophage formation and no phage transportation into the environment follows. At the same time the population of the infected cells reproduces like an ordinary polymorphic population in the HEC (i.e. according to the trophic strategy), acquiring immunity to that type of phage (Figure 6).

Evolution, Biodiversity and Ecology in Microbial Communities: Mathematical Modeling and Simulation with… http://dx.doi.org/10.5772/58302 121

**Figure 6.** Scheme of phage infestation.

*P* – population size;

120 Biodiversity - The Dynamic Balance of the Planet

– proportion of individuals having *dij* trait value in population (in the genetic spectra).

Intercellular substrates

*N1* … *Ni S1* … *Sj*

Synthesis strategy

( )

,...,,,...,

*ji*

Generate ODE system

To model a phage infection, the HEC provides extended objects of polymorphic populations: *polymorphic phage populations* and the *polymorphic population of infected cells* (the "normal"

The infection modeling includes the following phases: infection of the healthy cells through phage penetration from the environment into one part of the cell population, phage reproduction inside of the infected cells, and finally, the phages burst into the environ‐ ment after the lysis of cells. The infected cells form polymorphic populations, further reproduction of which may follow a lytic or a lysogenic pathway. The lytic pathway means death (lysis) of the infected cells with synchronous phage formation and their transporta‐ tion into the environment (the number of phages depends on their profusion). The lysogenic pathway means prophage formation and no phage transportation into the environment follows. At the same time the population of the infected cells reproduces like an ordinary polymorphic population in the HEC (i.e. according to the trophic strategy), acquiring

Calculate on [0, *Tmax*] *S1(Tmax) Sj*

*(Tmax)*

…

111

*SSNNf dt*

( )

,...,,,...,

*jij*

*jj jj*

== ==

)0(,...,)0( )0(,...,)0(

*SSSS NNNN SSNNf dt*

1 ,int1 ,int 1 ,int1 ,int 11

polymorphic populations are therefore regarded as "healthy").

 

 

=

=

1

*dS*

*j*

**Figure 5.** Common pattern of synthesis strategies calculation.

immunity to that type of phage (Figure 6).

*dS*

**2.4. Infection process modeling**

*Pj*

Assign initial substrate concentrations

The choice of the lysogenic or the lytic scenario depends on the conditions of the cell population at the moment of infection: optimum conditions lead to a lytic type, pessimal conditions– to a lysogenic. In the latter case, a part of the population randomly switches to the lytic form if conditions improve, causing the death of this part and phage generation (Figure 7).

The polymorphism of phages, the formation of new strains owing to mutations and competi‐ tion between strains are also described via the genetic spectra arithmetics. However, in the phage populations unlike prokaryotic populations, genes define a specific virulence (an ability to infect certain populations) and abundance (the number of copies per lysed cell).
