As development progresses, the number of cells that make up the embryo increases. Cell divisions (egg fragmentation) at the earliest stages of development occur evenly (synchronously). But in some species earlier, in others later, this synchrony is broken and the cells from which the rudiments of different organs are formed begin to divide with at different speeds. These differences in the rate of division can be considered as one of the first manifestations of their differentiation.

In mammalian embryos, already after the stage of 16–32 blastomeres, most of the cells begin to divide faster and form a trophoblast, the rudiment of the future placenta. The future embryo itself consists of only a few cells at these early stages. However, later in the course of development and growth, the embryo and then the fetus become many times larger than the placenta.

In amphibians at the blastula stage, consisting of several thousand cells, the future mesoderm makes up less than one third of all cells. But as development progresses, mesodermal derivatives - all muscles, almost the entire skeleton, circulatory system, kidneys, etc. - occupy at least 80% of the total mass of the tadpole.

Particularly visual uneven pace cell division in the morphogenesis of many invertebrates. In species with mosaic development, already at the stage of 30–60 cells, the rudiments of all major organs are identified and represented by very few cells (sometimes only two). Further, cell divisions in each rudiment are strictly programmed. For example, the early ascidian embryo contains 52 ectoderm cells, 10 endoderm cells and only 8 mesoderm cells. During subsequent development, the number of ectoderm cells increases by 16 times, endoderm by 20, and mesoderm by 50. Due to the programming of divisions, the number of cells in some adult invertebrates (for example, nematodes) is strictly constant and each organ is represented by a certain number of cells. The location of an organ and the place where its constituent cells divide do not always coincide. Often mitoses occur only in a special reproduction zone and from there the cells migrate to their place of differentiation. We have already seen examples of this kind when considering the stem cell system. The same thing happens, for example, during brain development.

The program of cell divisions is not always very strict and predetermines their exact number. More often, divisions probably occur until the number of cells or the size of the organ reaches a certain value. It's about Thus, about two fundamentally different mechanisms of regulation of cell divisions.

In one case (as in eggs with mosaic development), it is apparently contained in the dividing cell itself, which must “be able to count” its divisions. In another case, there must be some kind of “feedback loop” when the mass of an organ or the number of cells, reaching a certain value, begins to inhibit further divisions.

It turned out that the number of divisions in normal cells that are not transformed into malignant ones is not at all infinite and usually does not exceed 50–60 (most cells divide less, since if the egg were evenly divided 60 times, then the number of cells in the body (260) would be thousand times higher than in reality). However, neither the mechanism of such a limit on the number of cell divisions (called the Hayflick limit after the scientist who discovered it), nor its biological meaning is yet clear.

What is the “sensor” in the regulatory system – organ size or number of cells? An unambiguous answer to this question is provided by experiments with the production of animals with altered ploidy - haploid, triploid or tetraploid. Their cells are respectively 2 times smaller or 1.5 or 2 times larger than normal diploid ones. However, both the size of the animals themselves and the size of their organs are usually normal, that is, they contain more or fewer cells than normal. The controlled variable, therefore, is not the number of cells, but the mass of the organ or the entire organism.

The situation is different with plants. The cells of tetraploid plants, like those of animals, are correspondingly larger than diploid ones. But the sizes of parts of tetraploid plants - leaves, flowers, seeds - are often almost 2 times larger than usual. It seems that in plants the “sensor” for determining the number of cell divisions is not the size of the organ, but the number of cells itself.

The mechanisms regulating cell division - cell proliferation - are being studied very intensively and with different sides. One of the incentives for such activity by scientists is that the differences between cancer cells and normal cells largely consist in the disruption of the regulation of cell divisions, in the release of cells from such regulation.

An example of one of the mechanisms for regulating cell division is the behavior of cells seeded at the bottom of a bottle with a nutrient medium - a cell culture. In good conditions, their divisions occur until they cover the entire bottom and the cells touch each other. Next comes the so-called contact inhibition, or cell density-dependent inhibition. It can be disrupted, as Yu. M. Vasiliev did, by clearing a small window on the surface of the glass from cells. Cells rush into this window from all sides, and a wave of cell divisions passes around it. One might think that in the body, contacts with neighboring cells are a mechanism that restrains cell division.

In tumor cells, this regulation is disrupted - they do not obey contact inhibition, but continue to divide, piling up on top of each other. Unfortunately, they behave similarly in the body.

However, contact inhibition is not the only mechanism of regulation: its barrier can also be overcome in completely normal cells. For example, the liver cells of a young animal, tightly pressed together, nevertheless divide and the liver grows along with the growth of the entire animal. In adult animals, these divisions practically stop. However, if two lobes of the liver are removed, then massive cell divisions - liver regeneration - will very quickly begin in the remaining lobe. If one kidney is removed, within a few days the second kidney will double in size due to cell division. It is obvious that in the body there are mechanisms capable of stimulating cell division in an organ, activating its growth and thereby bringing the size of the organ into some quantitative correspondence with the size of the entire organism.

In this case, it is not contact mechanisms that operate, but some chemical factors, may be related to liver or kidney function. One can imagine that the insufficiency of the function of these organs, when part of them is removed or when their growth lags behind the growth of the entire organism, so disrupts the entire metabolism in the body that it causes a compensatory stimulation of cell divisions precisely in these organs. There are other hypotheses that explain, for example, such phenomena by the action of special inhibitors of cell division - keylons, secreted by the organ itself; if the organ is smaller, then there are fewer cells and more cell divisions in this organ. If such a mechanism exists, it does not operate everywhere. For example, the loss of one leg does not in itself lead to an increase in the size of the other leg.

The divisions of stem and differentiating blood cells are stimulated, as we have already said, by hormones, such as, for example, erythropoietin. Hormones stimulate cell division in many other cases. For example, stimulation of the growth of the number of oviduct cells in chickens is activated by the female sex hormone. There are chemical factors - usually these are small proteins that do not act like hormones, that is, they are not carried with the blood throughout the body, but have a more limited effect on neighboring tissues. These are now known growth factors - epidermal, etc. However, in most cases, specific chemical factors regulating cell division and the mechanisms of their action are unknown to us.

We know even less about the regulation of cell divisions during the main processes of morphogenesis - in embryonic development. We have already said that here the ability of some cells to divide faster than others is a manifestation of their differentiation. At the same time, one cannot help but notice that differentiation and cell division, in a certain sense, oppose each other and sometimes even exclude each other. In some cases, this is due to the impossibility of division during advanced, terminal differentiation of cells. Can, for example, a red blood cell with its very specialized structure, hard shell and almost complete loss of most cellular functions, and in mammals also with the loss of the nucleus? Although nerve cells maintain a very high metabolic rate, their long axon and dendrites connected to other cells serve as obvious obstacles to division. If such a division nerve cell did occur, it would result in the loss of that cell's communication with others and therefore the loss of its function.

Therefore, the usual sequence of events is first a period of cell proliferation, and only then differentiation, which is terminal in nature. Moreover, a number of scientists suggest that precisely during cell division, chromosomes are, as it were, “released” for the next stage of differentiation; the last mitosis before differentiation is given special importance. These ideas are still largely speculative and do not have good experimental foundations at the molecular level.

But even without knowing the specific mechanisms of regulation of cell divisions, we have the right to consider their programmed nature as the same manifestation of the development program as all its other processes.

In conclusion, we will briefly dwell on a phenomenon that seems to be the opposite of cell reproduction - their death, which in certain cases of morphogenesis is a necessary stage of development. For example, when fingers are formed in the rudiments of the hand of the fore and hind limbs, mesenchyme cells gather into dense cords, from which phalangeal cartilage is then formed. Among the cells remaining between them, mass death occurs, due to which the fingers are partially separated from each other. Something similar occurs during the differentiation of the wing primordium in birds. The mechanisms of cell death in these cases—factors external to the cells and events within the cells—remain poorly understood. A. S. Umansky suggests, for example, that cell death begins with the degradation of its DNA.

Cell reproduction, despite all its importance, cannot be considered the main mechanism of morphogenesis: it still participates indirectly in the creation of the form, although such important parameters as general shape organ and its relative sizes can be regulated precisely at the level of cell divisions. Programmed cell death plays an even smaller role in morphogenesis. But nevertheless they are in normal development absolutely necessary components. Almost all components of the cell and its genetic apparatus participate in the regulation of these phenomena. This shows us that in development there is no simple processes. An attempt to fully understand any of them forces us to turn to the basic molecular mechanisms of cell functioning. But there is still a lot unresolved here.

In order to appreciate the complexity of the development of a multicellular organism, one must imagine this process occurring as if in a multidimensional space. One axis is made up of a long chain of stages in the implementation of genetic information - from gene to trait. The second such axis can be called the entire set of genes in chromosomes. During development, the products of different genes interact with each other. The unfolding of events along two axes forms, as it were, a network on a plane. However, there is a third axis - the variety of events occurring in different parts embryo. These events can occur relatively autonomously, as in animals with mosaic development. But partially in them, and fully in species with a regulatory type of development, greater or lesser interactions and always complex cell movements occur between parts of the body. It is possible to consider them all as one axis only by making significant simplifications. And finally, all development (gametogenesis, embryogenesis and postembryonic development) occurs on a time scale that is completely different from the time measured along the path from gene to protein. Along this (conditionally fourth) axis, the entire multidimensional picture changes radically - the egg turns into a reproducing organism. This multidimensionality illustrates the complexity of all processes and their relationships and the difficulties of understanding them.

In some viruses, the role of hereditary substance is not performed by DNA, but by RNA, which is similar in structure.

§ 1. GENERAL PROCESSES OF PURE BIRTH (REPRODUCTION) AND POISSON PROCESSES

The previous chapters introduced basic concepts and discussed methods for analyzing discrete-time Markov chains. This chapter provides a brief discussion of some important examples of continuous-time discrete-state Markov processes.

More precisely, here we will deal with a family of random variables that take non-negative integer values. We will restrict ourselves to the case where the Markov process has stationary transition probabilities. Thus, the transition probability function at

does not depend on

Usually, when studying partial probabilistic models of physical phenomena, it is more natural to describe the so-called infinitesimal probabilities associated with the process, and then derive from them an exact expression for the transition function.

In the case under consideration, we will postulate a form for small ones using the Markov property, we will derive a system of differential equations that satisfies for all the solutions to these equations under the appropriate initial conditions. Recall that the Poisson process introduced in § 2 of Chap. 1 was considered in this way.

Before moving on to the general process of pure birth, let us briefly recall the axioms characterizing the Poisson process.

A. Postulates of the Poisson process

The Poisson process was considered in § 2 of Chapter. 1, where it was shown that it can be determined using a few simple postulates. In order to define more general processes of this kind, we point out some properties that the Poisson process has. Poisson process is

a Markov process that takes non-negative integer values ​​and has the following properties:

Property (1) can also be written this way.

Let's consider another typical scheme of continuous Markov chains - the so-called death and reproduction scheme, often encountered in a variety of practical problems.

Markov process with discrete states S 0 , S 1 , ..., S n called a process death and reproduction, if all states can be pulled into one chain in which each of the middle states ( S 1 , S 2 , ...,
S n -1) can only transition to neighboring states, which, in turn, transition back, and extreme states ( S0 and Sn) go only to neighboring states (Fig. 3.7).

The name is taken from biological problems, where the state of a population S k means the presence in it k units of individuals.

The transition to the right is associated with the reproduction of units, and the transition to the left is associated with their death.

Rice. 3.7. State graph for the process of death and reproduction

l 0 (t), l 1 (t), l 2 (t), …, l n (t)- reproduction intensity;

m 1 (t), m 2 (t), …, m n (t)- intensity of death.

U l And μ index of the state from which the arrow exits.

With a fortune S k not related random variable X k: if the system S at a point in time t is in a state S k, then the discrete random variable X(t), associated with the functioning of the system, takes on the value k. Thus, we obtain a random process X(t), which at random, previously unknown moments in time abruptly changes its state.

Markov process death and reproduction with continuous time is a random process that can only take on non-negative integer values. Changes in this process can occur at any point in time, that is, at any point in time it can either increase by one, or decrease by one, or remain unchanged.

In practice, processes of pure reproduction and pure death occur. A process of pure reproduction is a process of death and reproduction in which the intensities of all flows of death are equal to zero; Similarly, the process of pure “death” is a process of death and reproduction in which the intensities of all reproduction flows are equal to zero.

Example 1. Let's consider the operation of car models of the same brand in a large transport company (enterprise). The rate of arrival of cars to the enterprise is equal to l(t). Each car received by the enterprise is written off after a random time Tc. Vehicle service life t distributed according to the exponential law with the parameter m. The process of operating cars is a random process. A(t)- the number of cars of this brand in use at the time t. Let's find the one-dimensional distribution law of the random process P i (t) = P(A(t) = i), if: 1) there are no restrictions on the number of machines operated, 2) the enterprise can operate no more than n cars.

Solution.

1. The random process of car operation is a process of death and reproduction, the marked graph of which is presented in Fig. 3.8.

Rice. 3.8. State graph

The system of Kolmogorov equations corresponding to this graph has the form

Where i = 1, 2, …

If at the initial moment of time t= 0 there was not a single car at the enterprise, then this system of equations must be solved under initial conditions P0(0) = 1, P i (0) = 0 (i= 1, 2, …). If at t= 0 at the enterprise was k cars ( k= 1, 2, ...), then the initial conditions will have the form

P k (0) = 1, P i (0) = 0 (i = 1, 2, …, i¹k).

2. If the enterprise can operate no more than n cars of the same brand, then there is a process of death and reproduction with a limited number of states, the marked graph of which is presented in Fig. 3.9.

Rice. 3.9. State graph

The Kolmogorov system of equations for a labeled graph (Fig. 3.9) has the form (3.4).

This system must be solved under the initial conditions discussed above. Solutions of systems of equations (3.4) and (3.5) are one-dimensional distribution laws Р i (t). Finding solutions to systems in general view for an arbitrary function form l(t) presents significant difficulties and has no practical applications.

At constant intensities of death and reproduction flows and a finite number of states, a stationary regime will exist. System S with a finite number of states ( n+ 1), in which the process of death and reproduction occurs with constant intensities of death and reproduction flows, is the simplest ergodic system. The labeled state graph for such a system is shown in Fig. 3.9.

The limiting (final) probabilities of states for the simplest ergodic process of death and reproduction, which is in a stationary mode, are determined by the following formulas:

Rule. Probability k-th state in the scheme of death and reproduction is equal to a fraction, the numerator of which is the product of all reproduction intensities located to the left S k, and in the denominator is the product of all intensities of death located to the left S k, multiplied by the probability of the tap left state of the system P0.

In the previous example, for a stationary mode, if the rate of arrival of cars is constant ( l(t) = l = const), then the final probabilities of states, provided that there are no restrictions on the number of cars in the enterprise, are equal to

In this case, the mathematical expectation of the number of cars in use is equal to its variance:

M = D = l/m. (3.10)

If there is a limit on the number of cars at the enterprise (no more n), then the final probabilities can be written as follows:

Where ρ = l/m.

Where k = 0, 1, 2, ..., n.

Mathematical expectation of the number of vehicles in operation in stationary mode

Example 2. The production line includes four machines. A team of four maintenance personnel carries out preventive maintenance of each of them. The total flow of repair completion moments for the entire team is Poisson with intensity l(t). After the repair is completed, the machine is checked; with probability R it turns out to be efficient (the testing time is short and can be neglected compared to the prevention time). If the machine turns out to be inoperable, then its maintenance is carried out again (the time for which does not depend on whether it was carried out previously), etc. At the initial moment, all machines require preventive repairs. Required:

1. Construct a state graph for the system S(four machines).

2. Write differential equations for state probabilities.

3. Find the mathematical expectation of the number of machines Mt, the completion of those who have undergone prophylaxis by the time t.

Solution.

The state graph is shown in Fig. 3.10, in which:

S 0 – all four machines require preventive maintenance;

S 1– one machine has successfully completed preventive maintenance, and three require preventive repairs;

S 2– two machines have successfully completed preventative maintenance, and two require preventive repairs;

S 3– three machines have successfully completed preventive maintenance, one requires preventive repairs;

S 4– all four machines successfully completed preventative maintenance.

Rice. 3.10. System state graph

Each preventative repair ends successfully with the probability P, which is equivalent P- transformation of the flow of repair completions, after which it will remain Poisson, but with intensity Pl(t). In this example we are dealing with a pure reproduction process with a limited number of states.

Kolmogorov's equations have the following form:

Initial conditions P0(0) = 1, P 1 (0) = … = P 4 (0)= 0. At constant intensity l(t) = l and state probabilities are determined by the following formulas:

The mathematical expectation of the number of disks that have successfully completed maintenance by time t is equal to

Where n = 4.

Example 3. Consider the production of automobiles in a factory. The flow of produced cars is non-stationary Poisson with intensity l(t). Let's find the one-dimensional law of distribution of the random process X(t)- number of cars produced by time t, if at the moment t= 0 production of cars has begun.

Solution

Obviously, here is a process of pure reproduction without restrictions on the number of states, while l i (t) = l(t), since the intensity of car production does not depend on how many of them have already been produced. The state graph of such a process is shown in Fig. 3.11.

Rice. 3.11. State graph

One-dimensional distribution law of a random process X(t) for the graph shown in Fig. 3.11, is determined by the following system of Kolmogorov equations:

Since the number of cars produced X(t) at any fixed moment t distributed according to Poisson's law with parameter

M = D = a(t).

The process discussed in this example X(t) called inhomogeneous Poisson process. If intensity l(t) = l = const, then we get homogeneous Poisson process. For such a process at P0(0) = 1, P i (0) = 0 (i > 0)

The characteristics of the Poisson process will be

M = D = l×t.

Task 1. There is a device that consists of four units; The failure flow is the simplest, the average failure-free operation time of each node is 11 hours. The failed unit immediately begins to be repaired; The average repair time for a unit is 2 hours. (the recovery flow is the simplest). Find the average productivity of the device, if with four working nodes it is 100%, with three 60%, with two or less the device does not work at all.

One of the most important cases of Markov chains is known as the death and reproduction process. This process can be with discrete or continuous time, and its defining condition is that transitions only to neighboring states are allowed.

Let us consider the process of death and reproduction with continuous time. This process is a model of changes in population size.

The process is in the state To her, if the volume (number) of the population is equal to k; transition to state Ek corresponds to the death of one member of the population, and the transition to the state Ek+- birth.

This process can be considered as a QS model in which Ek corresponds To requests in the system, and the transition to the state Ek- or Ek+- the departure of an application from the system or its arrival.

For the process of death and reproduction with a set of states 0, 1,2, ..., the following conditions must be met:

Here P(+i; bt; k)- probability i births during bt provided that the population size is equal To; P(-i; bt; k)- probability i deaths under the same conditions.

According to these conditions, multiple births, multiple deaths, and simultaneous births and deaths within a short period of time are prohibited in the sense that the probability of these multiple events is of the order of smallness o(6r). This property follows from the property of exponential distribution, as shown earlier.

Let us find the probability that the population size at some point in time is equal to k p(k, t) = P.

Consider the change in population volume over a period of time (t, t+ 5/). At a moment in time t+bt the process will be in state E To, if one of three mutually exclusive and forming full group events:

• 1) at the moment of time t the population volume was equal to A: and during the time bt the condition has not changed;
• 2) at the moment of time t population size was To - 1 and per time bt one member of the population was born;
• 3) at a point in time t population size was To+ 1 and for time bt one member of the population died.

Then the probability that at time t+bt the process will be in the state Ek, equal to

The above equality makes sense only when to > Oh, because a population cannot consist of (-1) member. Boundary equality at To= O has the form:

In addition, the normalization condition must be satisfied

Isolating in equations (49.3) and (49.5) r(k) and dividing by bk we get

Going to the limit at bt-> 0, we have:

Thus, the probabilistic process under consideration is described by a system of linear differential equations. These equations can be derived directly from the state diagram (Figure 49.2).

Rice. 49.2.

State Ek indicated by an oval in which the number is written To. Transitions between states are indicated by arrows, which represent the intensities of the transitions.

The difference between the intensity with which the system enters the state Ek, and the intensity with which it leaves it must be equal to the intensity of the change in flow in this state.

Flow intensity per state

Flow intensity from state ~

The difference between them is equal to the effective intensity of the flow of probabilities into the state

It is impossible to solve this system in general form. The model of even a simple system is extremely complex and difficult to analyze. If we consider the QS more complex type, then the computational difficulties will be even higher. Therefore, solutions of system (49.3) - (49.4) are usually considered in a steady state at t-> oh, p"(k; t) -> 0,р(к, t) -> r(k)= const.

Process of pure reproduction

For this process p*=O, A* = A = const. It can be considered as a model of the flow of applications received by the QS. The system of equations for this process has the form:

Let the initial conditions be as follows:

Then and at k= 1 we get: exp

The solution to this equation is r(; /) = A/ exp (-AD By induction we can obtain that

Thus, the probabilities are distributed according to Poisson's law.

The Poisson process is central to QMS research. This is due, firstly, to its simplifying analytical and probabilistic properties; secondly, it describes many real processes that are a consequence of the cumulative effect large number individual events.

The simplest generalization of the Poisson process is obtained under the assumption that the probabilities of jumps can depend on the current state of the system. This brings us to the following requirements.

Postulates. (i) A direct transition from the state is possible only to the state . (ii) If at the moment of time the system is in the state , then the (conditional) probability of one jump in the subsequent short time interval between and is equal to whereas the (conditional) probability of more than one jump in this interval is .

Distinctive feature This assumption is that the time the system spends in any particular state plays no role; Sudden changes in state are possible, but as long as the system remains in the same state, it does not age.

Let again be the probability that at time the system is in state . These functions satisfy a system of differential equations, which can be derived using the arguments of the previous paragraph with the only change that (5) in the previous paragraph is replaced by

Thus, we obtain the basic system of differential equations

In a Poisson process, it was natural to assume that at time 0 the system leaves the initial state. We can now allow for a more general case where the system leaves an arbitrary initial state. Then we get that

These initial conditions uniquely determine the solution of system (2). (In particular, ). Explicit formulas for were derived independently by many authors, but they are not of interest to us.

Example. Radioactive decay. As a result of the emission of particles or -rays, a radioactive atom, say uranium, can turn into an atom of another type. Each kind represents a possible state, and as the process proceeds we get a sequence of transitions. According to accepted physical theories, the transition probability remains unchanged while the atom is in the state, and this hypothesis finds expression in our initial assumption. Therefore, this process is described differential equations(2) (a fact well known to physicists). If is a final state from which no other transitions are possible, then system (2) terminates at . (When we automatically receive ).