Mathematical and Computer Modelling 44 (2006) 595–607
www.elsevier.com/locate/mcm
Stochastic mechanics in the context of the properties of
living systems
E. Mamontov a,∗ , K. Psiuk-Maksymowicz a , A. Koptioug b
a Department of Physics, Göteborg University, 412 96 Göteborg, Sweden
b Department of Information Technology and Media, Mid Sweden University, 831 25 Östersund, Sweden
Received 24 January 2006; accepted 30 January 2006
Abstract
Many features of living systems prevent the application of fundamental statistical mechanics (FSM) to study such systems.
The present work focuses on some of these features. After discussing all the basic approaches of FSM, the work formulates an
extension of the kinetic theory paradigm (based on the reduced one-particle distribution function) that exhibits all of the livingsystem properties considered. This extension appears to be a model within the generalized kinetic theory developed by N. Bellomo
and his co-authors. In connection with this model, the work also stresses some other features necessary for making the model
relevant to living systems. A mathematical formulation of homeorhesis is also derived. An example discussed in the work is a
generalized kinetic equation coupled with a probability-density equation representing the varying component content of a living
system. The work also suggests a few directions for future research.
c 2006 Elsevier Ltd. All rights reserved.
Keywords: Fundamental statistical mechanics; Living system; Homeorhesis; Generalized kinetic theory; Stochastic process
1. Introduction
The minimal unit of life is a living cell. A system which does not include living cells can be termed a nonliving
system. Nonliving systems are studied in most of areas of physics, chemistry, and engineering. These fields are
endowed with a great variety of experimental methods and theoretical treatments, including fundamental statistical
mechanics (FSM; e.g., [1–4]). The latter proved to be a powerful tool in the analysis of problems in nonliving matter.
Subsequently, there was an irresistible temptation to apply the first-principles mechanical theory to virtually every
new problem, in particular, to those living-system problems that appear similar to the ones successfully resolved by
the nonliving-matter sciences in the previous decades. Thus, one can expect that physicists would suggest FSM as a
unified theory suitable for addressing both nonliving and living systems.
∗ Corresponding author.
E-mail addresses: [email protected] (E. Mamontov), [email protected] (K. Psiuk-Maksymowicz),
[email protected] (A. Koptioug).
c 2006 Elsevier Ltd. All rights reserved.
0895-7177/$ - see front matter doi:10.1016/j.mcm.2006.01.028
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E. Mamontov et al. / Mathematical and Computer Modelling 44 (2006) 595–607
Alongside this, there are other opinions. One of them comes from biology [5, p. C49]:
Biological systems are very different from the physical or chemical systems analysed by statistical mechanics or
hydrodynamics. Statistical mechanics typically deals with systems containing many copies of a few interacting
components, whereas cells contain from a million to a few copies of each of thousands of different components,
each with very specific interactions.
The authors of this continue their thought in the following way:
. . . The macroscopic signals that a cell receives from its environment can influence which genes it expresses —
and thus which proteins it contains at any given time – or even the rate of mutation of its DNA, which could lead
to changes in the molecular structures of the proteins. This is in contrast to physical systems where, typically,
macroscopic perturbations or higher-level structures do not modify the structure of the molecular components.
The present work analyzes in what specific respect FSM does not allow for the above features and what stochastic
mechanics instead of FSM can enable them. The work singles out the version of stochastic mechanics most relevant,
among available alternatives, to the study of living systems.
Biologists emphasize many features of living systems which do not allow one to apply FSM to study these systems
(see Sections 1 and 2.1). The results of the present work are the following. After discussing all the basic approaches of
FSM (see Sections 2.2–2.6), the work formulates (see Section 2.6) an extension of the kinetic theory paradigm (based
on the reduced one-particle distribution function) that possesses all of the living-system properties considered in the
work. This extension appears to be a model within the generalized kinetic theory developed by N. Bellomo and his
co-authors (e.g., [6–8]; see also [9–12]). In connection with this model, some other features necessary for making the
model relevant to living systems are also stressed (see Section 2.7 and the Appendix). An example discussed in this
work (see Section 3) is a generalized kinetic equation coupled with a probability-density equation which represents
the varying component content of a living system. The work also suggests a few directions for future research. One of
them is accounting for homeorhesis, the key phenomenon of the living-system dynamics. A mathematical formulation
of homeorhesis is derived in terms of a generic dynamical treatment, namely ordinary differential equations (see the
Appendix).
2. Analysis
2.1. Implications from biology: Formulation in terms of FSM
The issues in the two quotations in Section 1 can be formulated in terms of FSM. We consider a population of
particles which occupies a bounded or unbounded set X (t) ⊆ R3 where R = (−∞, ∞) and t ∈ R is the time. For all
t, set X (t) is assumed to be a domain, i.e. a set which is connected and open in the topology in R3 . To apply various
treatments below, it is assumed that, if X (t) 6≡ R3 , then the boundary of X (t) is piecewise smooth and the conditions
at this boundary are available but generally depend on the solutions of the models employed. Also note that domain
X (t) with the above properties is a Borel set.
Let the population comprise
M ≥1
(1)
components, i.e. (different if M 6= 1) groups of identical particles, and let Ni , such that
i = 1, . . . , M,
Ni ≥ 1,
(2)
be the number of the particles in the ith component. The total number N of the particles in the population is expressed
as
N=
M
X
Ni .
(3)
i=1
Obviously,
N ≥ M.
(4)
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597
Equality N = M corresponds to the case when every component comprises exactly one particle or, equivalently, all
the particles are pairwise different. This can take place in a living system.
Remark 1. It follows from the first quotation in Section 1 that M and Ni in (1) and (2) are not limited to any specific
intervals, i.e.
(A) number M in (1) can take any value from very low, such as unit or a few units, to a few millions (or greater),
(B) every number Ni in (2) can take any value from very low, such as unit or a few units, to a few thousands (or
greater).
These are important properties of a living-particle population.
The second quotation in Section 1 implies that, generally speaking, the numbers in Remark 1 depend on time t, i.e.
(5)
M = M(t)
and
Ni = Ni (t),
i = 1, . . . , M.
(6)
Substituting (6) into (3) one obtains
N = N (t)
(7)
which also allows for (5). The four sections below analyze whether any specific model within FSM allows the
properties in Remark 1 and features (5), (6).
2.2. Phase-space models
In FSM, each particle is described with the three-dimensional position vector in the domain X (t) (see the beginning
of Section 2.1) and with the three-dimensional momentum vector. Thus, the phase space for one particle is R6 .
Subsequently, the phase-space models in FSM consider the N -particle population in the phase space Rd , where
d = 6N .
The corresponding key notion is the phase-space probability density (e.g., [3, (3.7), (3.8), and p. 25])
ϕ(N , t, x1 , p1 , . . . , x N , p N )
(8)
where variables x j ∈ X (t) and p j ∈ R3 represent the position and momentum vectors, respectively, of the jth particle.
Density (8), as a function of x j and p j , j = 1, . . . , N , at every fixed N and t, is also the joint probability density of
the positions and momenta of N particles at time t.
The phase-space models form the core of FSM. They include the Liouville equation (e.g., [3, (3.15)]), the formal
master equation (e.g., [2, (VIII.2), (VIII.21)]), the molecular dynamics equation systems, and other formalisms. Along
with this, it is unclear how to use the phase-space settings under condition (7). Generally speaking, this prevents
application of the phase-space models to living systems. To resolve the problem, one chooses a phase space such that
its dimension is independent of varying number (7). The simplest way to do that is discussed in Section 2.3.
2.3. One-particle reduced distribution function: One-component population
The simplest way to make the phase-space dimension independent of N (t) (cf., (7)) is by using the six-dimensional
space for one particle as the phase space for the entire population. In this respect, the 6N (t)-dimensional space
is reduced to the 6-dimensional one. However, the reduction can be done only under a special condition. This is
considered below.
Joint probability density (8) enables one to introduce functions ρ j with equalities
ρ j (N (t), t, x j , p j ) =
j = 1, . . . , N (t).
Z
[X (t)×R3 ] N (t)−1
ϕ(N (t), t, x1 , p1 , . . . , x N (t) , p N (t) )
N
(t)
Y
dxk d pk ,
k=1,k6= j
(9)
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If density function ϕ at every fixed t is symmetric with respect to all pairs (xk , pk ) (e.g., [3, (4.7)]) or, because of (9),
ρ1 (N (t), t, x, p) ≡ ρ2 (N (t), t, x, p) ≡ . . . ≡ ρ N (t) (N (t), t, x, p),
(10)
then the N (t)-particle population can be described with the reduced one-particle distribution function (e.g., [3, (4.9)])
f 1 (t, x, p) = N1 (t)ρ1 (N1 (t), t, x, p)
(11)
where identity
N (t) ≡ N1 (t)
(12)
holds since it, in view of (3) and (7), is equivalent to
M =1
(13)
which, in turn, is equivalent to condition (10).
In kinetic theory, distribution function f 1 in (11) is granted as a solution of a kinetic equation (e.g., [3, p. 55]). In
so doing, the terms on the right-hand side of (11) are determined by means of this function as follows
Z
N1 (t) =
f 1 (t, x, p) dx d p,
ρ1 (N1 (t), t, x, p) = f 1 (t, x, p)/N1 (t).
(14)
X (t)×R3
The condition that all the particles are identical disagrees with living-systems feature (A) of Remark 1 and is
eliminated in Section 2.5. The next section discusses why the most important extension of the above paradigm is
also unsuitable for the living-systems modelling.
2.4. Two-particle and multiparticle reduced distribution functions: One-component population
This section considers the particle population which is one-component, i.e. the case when (12) or (13) holds. The
one-component settings in the previous section can be regarded as a quite particular case of a more refined treatment,
the Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) equation chain (e.g., [3, p. 45]). It is based on the reduced
l-particle distribution functions, l ≥ 1. The case in Section 2.3 corresponds to l = 1. The case when l > 1 is discussed
in the present section.
The well-known normalization condition for the reduced l-particle distribution function at l > 1 (e.g., [3, (4.14)
and (4.18)]) is equivalent to the relation (e.g., [3, p. 41])
N (t)[N (t) − 1] · · · [N (t) − l + 1] ≈ [N (t)]l ,
N (t) ≥ l > 1.
(15)
One can show that the relative error of this approximation is inversely proportional to N (t),
∼ [N (t)]−1 .
(16)
Since the BBGKY description is traditionally regarded as a first-principles model, approximate equality (15) should
hold with a very low relative error , say, on the order of tenths or hundredths of one per cent, i.e. = 10−3 –10−4 .
Application of these values to (16) requires N (t) ∼ 103 –104 .
This limitation sharply disagrees with the living-system feature (B) of Remark 1 (see also (12)). Subsequently, the
reduced l-particle distribution functions at l > 1 and any model employing at least one of them (e.g., the BBGKY
equation chain or the so-called correlation functions (e.g., [3, Section 4.4])) are in general not suitable for describing
living systems. Subsequently, one has to come back to the reduced one-particle distribution function in Section 2.3.
2.5. One-particle reduced distribution function: Multicomponent population
The common extension of the identical-particle treatment in Section 2.3 to the case when the particles do not need
to be identical involves the notion of a multicomponent population. Indeed, no matter whether M = 1 or M > 1, one
can without loss of generality assume that the first M particles in the N (t)-particle population (cf., (4)) are pairwise
different and any other particle is identical to one of the previously mentioned M particles. In so doing, number Ni (t)
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599
in (2) and (6) turns out to be the total number of the particles identical to the above ith particle where i = 1, . . . , M.
The corresponding generalization of (11) is
f (t, x, p) =
M
X
f i (t, x, p)
(17)
i=1
where
f i (t, x, p) = Ni (t)ρi (Ni (t), t, x, p),
i = 1, . . . , M
(18)
and f i are the reduced one-particle distribution functions for the components.
In kinetic theory, distribution functions f 1 , . . . , f M in (18) are obtained as a solution of a system of M kinetic
equations (e.g., [4, Section 3.2.2]). Note, however, that the derivation of this system from representation (17) is not
always included in the corresponding textbooks. Solutions of the system are in set [0, ∞) M . Applying (5) to this set,
one encounters difficulties analogous to those described in Section 2.2. Thus, the above multicomponent treatment
underlying (17) is generally not suitable for living-systems modelling.
The problem is overcome in the next section by leaving the scope of FSM.
2.6. One-particle reduced distribution function: Time-dependent number of components. Generalized distribution
function
As it is stressed above, the notion of a multicomponent population is inherently associated with the features of the
particles of being either identical or different. The latter is resolved by comparing the component-specific values of
the related parameters of the particles, for instance, the particle mass, size, shape, electric charge, and other physical
quantities. They can be regarded as the entries of a vector variable, say, u ∈ Rm (where m ≥ 1) such that
• the ith component, i = 1, . . . , M, corresponds to the value u i of u and
• all the values u 1 , . . . , u M ∈ Rm are pairwise different, i.e.
u i 6= u j ,
i 6= j, i, j = 1, . . . , M.
(19)
The treatment below follows the single-distribution-function approach to multicomponent populations which is
proposed in ([10,9]). According to it, M functions f i (t, x, p) in (18) can be regarded as M values f (t, x, p, u i ) of
the single function f (t, x, p, u) where vector u varies in a bounded or unbounded, generally time-dependent domain
in Rm , say, U (t) ⊆ Rm . This transforms (17) into
f (t, x, p) =
M
X
f (t, x, p, u i ).
(20)
i=1
In so doing, functions Ni and ρi in (18) are also determined in terms of the function f (t, x, p, u), namely
Ni (t) = N (t, u i ),
ρi (Ni (t), t, x, p) = ρ(N (t, u i ), t, x, p, u i ),
i = 1, . . . , M,
(21)
where (cf., (14))
N (t, u) =
Z
X (t)×R3
f (t, x, p, u) dx d p,
ρ(N (t, u), t, x, p, u) = f (t, x, p, u)/N (t, u).
(22)
(23)
Thus, the reduced one-particle distribution function f (see (20)) for the multicomponent population and other
basic characteristics of the components, for instance, (21)–(23), are completely described by means of the single
distribution function f . The latter can be determined in the way discussed in the next section. Now we shall analyze
how distribution function f enables one to allow for the living-system feature (5).
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The above set {u 1 , . . . , u M } can be regarded as a sample for a random variable defined on domain U (t) and
described with a probability density, say, λ(t, ·). In the simplest case, i.e. when the random variable is defined on Rm
and discrete, namely
λ(t, u) ≡
M
X
M −1 δ(u − u i ),
u ∈ U (t), U (t) ≡ Rm ,
(24)
i=1
where
δ(·) is the M-dimensional Dirac delta function, expression (20) is equivalent to f (t, x, p) =
R
M U (t) f (t, x, p, u)λ(t, u) du.
According to the approach of [10,9], one applies the latter equality even when (5) holds and both domain U (t) and
probability density λ(t, ·) are not necessarily of the particular form shown in (24), i.e.
Z
f (t, x, p, u)λ(t, u) du.
(25)
f (t, x, p) = M(λ(t, ·))
U (t)
In this way, the component number M(λ(t, ·)) specifies the t-dependence (5). The time-dependent component number
can be interpreted in various ways. The discussions in ([10, Sections 5 and 7], [9, Section 4.3 and Appendix]) suggest
reading it as the number of the modes of probability density λ(t, ·). These modes may be viewed as the spread out,
nonzero-width and finite-height peaks analogous to those in (24). Along with this, we present a more general and
somewhat more precise definition of M(λ(t, ·)).
Definition 1. Let the components of a population of particles be described with vector parameter u ∈ Rm . Let also
the number of these components be denoted by M(λ(t, ·)). We term any path-connected subset of set
Υ (λ(t, ·)) = {u ∈ Rm : λ(t, u) > 0}
(26)
the component-parameter subset of the population. The particles in a component of the population are the ones
corresponding to the values of u in a component-parameter subset. Subsequently, number M(λ(t, ·)) is the number of
the component-parameter sets in set (26).
If set (26) includes more than one component-parameter subset, then, obviously, all these subsets are mutually nonintersecting. This feature generalizes condition (19) which is valid in the case of (24) when the component-parameter
subsets are single-point sets {u 1 }, . . . , {u M }.
Importantly, Definition 1 is more general than the mode-based definitions developed in [10,9]. Indeed, any
collection of the population particles which is a component in the sense of the present definition comprises at least
one collection which is a component in the sense of the previous definitions.
Remark 2. As is well known [13, p. 1306] life is “the principle or force by which animals and plants are maintained
in the performance of their functions and which distinguishes by its presence animate from inanimate matter; the state
of a material complex or individual characterized by the capacity to perform certain functional activities including
metabolism, growth, reproduction, and some form responsiveness or adaptability”. Probability density λ in (25) and
its component characteristics in Definition 1 are the very quantities which can be the core in the representing of the
above metabolism, growth, reproduction, and responsiveness or adaptability.
Expression (25) shows that the entire multicomponent population is modeled in terms of a single, generalized
reduced one-particle distribution function
f (t, x, p, u) = M(λ(t, ·)) f (t, x, p, u)λ(t, u)
(27)
that describes not only stochastic position and momentum of a particle (represented with x and p, respectively)
but also its stochastic parameter vector (represented with u ∈ U (t)) which determines the particle content of the
population. In the context of (25) and (27), f is the conditional distribution function for (x, p) conditioned with value
u, f (see (25)) is the marginal distribution function for (x, p), ρ (see (23)) is the conditional probability density
for (x, p) conditioned with the values u and N (t, u) (see (22)), and λ is the marginal probability density for u.
Generalized distribution function (GDF) (27) is of the type underlying the generalized kinetic (GK) theory developed
by N. Bellomo and co-authors (e.g., [6–8]; see also [9–12]).
All the aforementioned issues can be summarized in the following way.
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Remark 3. Definition 1, single generalized reduced one-particle distribution function (27) and related expressions
(21)–(23), (25) enable one to take into account not only the features in Remark 1 and (6), but also (5), i.e. all the
properties of living systems discussed. Among the stochastic mechanics models considered in the present work,
representation (27) and the related expressions are the only treatment that can allow for the features in Remark 1
as well as (5) and (6). Subsequently, if a population of living particles can be modeled by a stochastic mechanics
approach, then the latter is based on (27). In other words, application of GDF (27) is the necessary condition for
modelling living systems by stochastic mechanics.
The next section derives one more condition that the model in Remark 3 should meet to properly describe the
mechanics of the living-particle population.
2.7. Some other properties of living system formulated in terms of the present model
Stochastic variables (x, p) and u represent the mechanics and component content, respectively, of a population of
living particles. The mechanical evolution described with f can be recognized as that of a living system only if the
properties of the particles (e.g., those in Definition 1) influence the above evolution. This means that f must depend
on λ,
f (t, x, p, u) = f (t, x, p, u, λ).
(28)
The corresponding correction should be taken into account in (22), (23), (25) and (27).
Because of homeorhesis [14, p. 32] (see also [15, p. 526] and [16]), GDF f (see (27)) cannot be stationary, i.e.
independent of t, in a system which is living. Along with this, a living system may in principle be mechanically
stationary, i.e. t-independence of f may be the case, at least as an approximation. In view of this and (27), probability
density λ is always nonstationary, i.e. depends on t. Paper [17] discusses the corresponding examples of experimental
results. A mathematical reading of homeorhesis is developed in the Appendix. Other details on the phenomenon in
connection with (27) can be found in ([9, Section 4.1]).
Numerous observations of mature (non-embryonic) living systems show that, in any sufficiently short time
intervals, living systems behave similarly to nonliving bodies, namely those with t-independent λ. For instance, if
the time needed for a driver to stop a car avoiding collision with an immobile obstacle is less than one second (the
typical time for the driver reaction), then the collision is inevitable. In a short time interval when the mechanical
dynamics function f noticeably varies, the driver action associated with λ remains unchanged. In specific terms, this
means that, if a mature living system is modeled with (27), then
1 τλ (u)/τ f (u) < ∞,
u ∈ U (t) ∩ Υ (λ(t, ·)),
(29)
where set Υ (λ(t, ·)) is described with (26), τλ (u) is the characteristic time of changes in λ(·, u), τ f (u) =
sup(x, p) {τ (x, p, u)}, and τ (x, p, u) are the characteristic times of changes in f (·, x, p, u, λ) (see (28)). The time
scales in living systems are discussed, for instance, in ([9, Section 4.4]).
The above issues can be summarized as follows.
Remark 4. If a living system is adequately described with the model in Remark 3, then:
• conditional distribution function f depends on probability density λ, i.e. (28) holds;
• probability density λ depends on t;
• inequality (29) is valid.
Moreover, the key phenomenon in the dynamics of a living system is homeorhesis. Its mathematical formulation
in terms of ordinary differential equations (in Euclidean space or a function Banach space) is derived in the
Appendix.
If any of the above conditions does not hold, then the system is nonliving.
The above summary lists the features to be accounted for in the development of specific models for GDF (28).
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3. Generalized kinetic equation and related models. Directions for future research
Conditional distribution function f (t, ·, ·, u, λ) in (27) and (28) can be obtained as a solution of the following
generalized kinetic equation (GKE) [9, (3.15)]
3
3
X
∂ f (t, x, p, u, λ) X
∂ f (t, x, p, u, λ) pi
∂ f (t, x, p, u, λ)
+
+
Fi [ f ](t, x, p, u, λ)
∂t
∂ xi
µ(u) i=1
∂ pi
i=1
Z
J2 [ f ](t, x, p, λ; u, u 0 )λ(t, u 0 ) du 0 ,
= J1 [ f ](t, x, p, λ; u) + M(λ(t, ·))
U (t)
x ∈ X (t), p ∈ R , u ∈ U (t) ∩ Υ (λ(t, ·))
3
(30)
where scalar µ(u) is the mass of a particle in the population component corresponding to u, and vector
F[ f ](t, x, p, u, λ) is the force acting on a particle and originating from phenomena other than collisions of the
particles with the surroundings or each other.
The term J1 [ f ](t, x, p, λ; u) is the collision integral due to the collisions of the particles of one component, i.e. the
component corresponding to u, with the surroundings. The term J2 [ f ](t, x, p, λ; u, u 0 ) is the collision integral due to
the collisions of the particles of two different components, i.e. the ones corresponding to u and u 0 . Specific examples
of the collision integrals with a detailed discussion can be found, for instance, in [2, Section 3 of Chapter IX] or [3,
Sections 7.2 and 7.3].
The collisions of three and more components are not included in GKE (30). Further details on this equation can be
found in ([9, Section 3.3]). We only note that, in case of (24), GKE (30) becomes a common kinetic equation system
for a multicomponent particle population (cf., [4, Section 3.2.2]).
Remark 5. GKE (30) is an example of a model that can provide the first feature of the living-system mechanics
discussed in Remark 4.
Note that, in the GK theory, conditional distribution function f (t, ·, ·, u, λ) may depend not only on the entries of
the component-content vector u but also on other scalar parameters. The latter can be determined by means of the
methods of the GK theory (see the references in the text above Remark 3).
New effects can be accounted for in GKE (30) by incorporating into the right-hand side a detailed treatment of
many-body collisions in multicomponent populations of living particles. This presumes a detailed reading of the
terms on the right-hand side of GKE (30) and complementing them with the corresponding extra terms. A possible
path in this direction is discussed in [18,19].
The general form of the model for probability density λ is the equation [9, (3.16)]
∂λ(t, u)/∂t = Λ(t, u, f (t, ·, ·, ·, λ), λ(t, ·)),
u ∈ U (t).
(31)
Eqs. (30) and (31) form the equation system for f (t, ·, ·, u, λ) and λ(t, u). Solutions of this system supposedly have
the second and third properties of the living-systems mechanics referred to in Remark 4. These solutions should also
comply with the mathematical implications from homeorhesis derived in the Appendix.
It follows from feature (A) of Remark 1 that Eq. (31) can be fairly complex. The fact that (31) must be regarded in
conjunction with GKE (30) to make the model closed fully agrees with well-known issues [5, p. C49]:
. . . the components of physical systems are often simple entities, whereas in biology each of the components is
often a microscopic device in itself, able to transduce energy and work far from equilibrium. As a result, the
microscopic description of the biological system is inevitably more lengthy than that of a physical system, and
must remain so, unless one moves to a higher level of analysis.
The form of function Λ in (31) is still unknown. In spite of the fact that the model of [20–22] takes into account
certain crucial properties of living systems (see the text from (A.12) down to the end of the Appendix), this model does
not allow for (5) and employs the simplest version (24) of Λ. In fact, it in part applies the nonliving-system paradigm.
More research is needed to specify function Λ. One of the fields for future study is associated with the fact that λ is
assigned to represent the component structure due to biochemical reactions in the particle population. Subsequently,
the results on the complex nature of these reactions (e.g., [23]) may contribute to derivations of function Λ.
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603
A discussion on equation (31) is also included in [9, Section 3.4]. We only add a few notes.
One of the simplest cases of equation (31) is the Kolmogorov forward, or Fokker–Planck, (KFFP) equation (e.g.,
[24] and the references therein). Research on the KFFP reading of (31) can be facilitated by a series of recent results.
For instance, the KFFP-based nonstationary invariant probability densities may appropriately describe homeorhesis
(cf., the text below (28); see also the Appendix). A fairly general treatment for these densities is reported in [25]. The
work [24] includes a group of analytical–numerical methods developed for KFFPs with nonlinear coefficients in the
high-dimensional case, i.e. when dimension m of vector u in (31) is high.
Acknowledgements
The present work was partly supported by the Instituto Nazionale di Alta Matematica Francesco Severi
(Rome, Italy). The authors thank the European Commission Marie Curie Research Training Network MRTNCT-2004-503661 “Modelling, Mathematical Methods and Computer Simulation of Tumour Growth and Therapy”
(http://calvino.polito.it/˜mcrtn/) for the full support of the second author. The authors are very grateful to N. Bellomo,
Politecnico di Torino (Turin, Italy), for a series of stimulating and fruitful discussions during the research stay of the
first author at Politecnico di Torino in autumn 2005.
Appendix. Dynamics of a living system in terms of ordinary differential equations
The main distinguishing features of the dynamical behavior of a living systems have been well known in biology
since long ago [14, Chapter 2], [15] (see also [16] for a recent discussion). However, they have not been formulated in
the language of mathematics yet. The present appendix derives the corresponding mathematical formulations in terms
of ordinary differential equations (ODEs).
The aforementioned features are the following.
(i) A living system is an open system.
(ii) A living system develops along a trajectory (in the space of the system states) that is completely determined by
the internal properties of the system and its environment, i.e. the related exogenous signals.
(iii) The trajectory called the creode (the necessary route or path) [14, p. 32] corresponds to the “most favored”
exogenous signals [14, p. 19]. The latter are also known as the formative drives (according to Aristotle’s theory
of epigenesis).
(iv) The actual trajectory, i.e. the one corresponding to the actual rather than “most favored” exogenous signals, need
not be the creode, but in the course of time tends to the latter, no matter what the signals are (in a certain, systemrelevant range). This “stability” with respect to the exogenous signals is known as homeorhesis [14, p. 32].
Homeorhesis is the time-dependent generalization of homeostasis. The latter term was coined by W.B. Cannon
in 1926 who discussed homeostasis in detail later [26].
Remark A.1. In the behavior of any living system, one recognizes a purposeful component, however, without
indicating the corresponding “mechanism” or specifying the terms. The resolution of the latter problem is substantially
contributed by Waddington’s theory of homeorhesis. Indeed, it follows from feature (iv) above that the homeorhetic
“mechanism” indicates the creode trajectory (which is independent of the actual exogenous signal) as the “purpose”
of a living system. This purpose is not invariable: its evolution is described with the creode dynamics.
Remark A.2. Exogenous signals are parameters of the environment of a living system but are not parts of the system.
They vary in time smoothly and are independent of the moments t B and t D (t B < t D ) of the birth and death of the
system.
We denote an exogenous signal with s(t) and regard it as a vector in the k-dimensional Euclidean space, i.e.
s(t) ∈ Rk where k ≥ 1. In view of Remark A.2, s ∈ Bk where Bk is the set of the functions which have values in Rk ,
and are defined, sufficiently smooth and uniformly bounded on the entire time axis R.
Waddington (see [14, p. 22]) suggested regarding the state space of a living system as Euclidean space, say, Rn
(n ≥ 1) and describing the system with an ODE of a fairly general form in Rn . The coordinates in the state space
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E. Mamontov et al. / Mathematical and Computer Modelling 44 (2006) 595–607
can include, for instance, concentrations (or volumetric number densities) of cells or molecules [14, Chapter 2 and
Appendix]. We denote by ν a state vector of a system which may be living and describe the system with the ODE
dν/dt = h(ν, w),
w = s(t)
(A.1)
where vector w ∈ Rk is the variable which represents the values of exogenous signals and function h has common
features, in particular, it is defined and sufficiently smooth in Rn+k . Moreover, we assume that every solution of (A.1)
exists for every t ∈ R.
The entire time axis R comprises three intervals where the system behaves differently. In intervals (−∞, t B ] and
[t D , ∞), i.e. before the birth and after the death, the system is not living. It is living in the life interval (t B , t D ), so t D −
t B > 0 is the system life span. Function h in (A.1) is related to the nature of the living system whereas the environment
is represented with function s. In view of these issues and feature (ii), any trajectory of the living system and, thus, its
birth and death moments t B and t D must be determined solely in terms of functions h and s, i.e. as a special solution
of ODE (A.1), without involving any initial condition. This leads to the mathematical property formulated as follows.
Property A.1. Eq. (A.1) at any s ∈ Bk has a unique solution in Bn , say, Ψ (t, s), i.e.
dΨ (t, s)/dt = h(Ψ (t, s), s(t))
(A.2)
and Ψ (·, s) ∈ Bn . This solution describes the only system trajectory corresponding to exogenous signal s(t), no
matter whether the system is living or nonliving. The trajectory also defines the birth and death moments t B and t D
because of feature (ii) and the fact that they depend solely on the exogenous signals and the nature of the system.
Feature (i) and Remark A.2 imply another assumption.
Property A.2. If Property A.1 holds, then, at any s ∈ Bk , solution Ψ (t, s) depends on t if the system is living, i.e. in
the life interval (t B , t D ).
Remark A.3. Property A.2 remains true if functions s are independent of t, i.e. ODE (A.1) is autonomous. In this
specific respect, solution Ψ (t, s) of autonomous ODE (A.1) in interval (t B , t D ) can be regarded as “self-oscillating”
in a broad sense. In particular, it may be periodic, quasi-periodic, or almost periodic. Living-system behaviors of this
kind were mentioned by Waddington (see [14, p. 33]).
We denote the “most favored” exogenous signal mentioned in feature (iii) as sχ (t). It need not coincide with actual
exogenous signal s(t) and hence is, generally speaking, hypothetical. (As explained in the text below (A.11), it is
“designed” by the living system itself.)
In view of feature (iii) and Property A.1, the creode is trajectory Ψ (t, sχ ). It is the solution of ODE (A.1) under the
condition that s ≡ sχ , i.e.
dΨ (t, sχ )/dt = h(Ψ (t, sχ ), sχ (t)).
(A.3)
The above issues lead to the following formulation of feature (iv) in the terms based on ODE (A.1).
Remark A.4. Let the living system be considered only in a subinterval of the life interval (t B , t D ) located sufficiently
far from both t B and t D , i.e. where the life fully performs. In this subinterval, the effects of both the birth and death
moments t B and t D on any solution Ψ (t, s) (which has Property A.1) can be approximately neglected. As a result, the
living state Ψ (t, s) can be regarded as the steady state, i.e. the one defined on the entire axis R rather than in a finite
interval (t B , t D ). This allows one to study asymptotic behavior by means of precise relations thereby avoiding the use
of semi-qualitative inequalities (such as “much less” or “much greater”).
Within the above steady-state approximation, one can mathematically formulate homeorhesis (see feature (iv)) as
follows. For every sufficiently small ks − sχ k, when t → ∞ vector Ψ (t, s) − Ψ (t, sχ ) is independent of time and,
moreover, is zero. Relations
d[Ψ (t, s) − Ψ (t, sχ )]
=0
dt
lim [Ψ (t, s) − Ψ (t, sχ )] = 0
lim
t→∞
t→∞
more precisely express the above-mentioned limit behavior.
(A.4)
(A.5)
E. Mamontov et al. / Mathematical and Computer Modelling 44 (2006) 595–607
605
It is still unclear in what specific respect exogenous signal sχ involved in (A.5) is the “most favored” one. To
elucidate that, one has to develop the corresponding mathematical formulation. The latter can be done in conjunction
with feature (iv) as described in the theorem below.
Theorem A.1. Let the assumptions mentioned in the text above Property A.1 and Property A.1 itself be valid. Let
also functions s and sχ be such that ks − sχ k is sufficiently small. We also assume that the steady-state approximation
introduced in Remark A.4 is the case and the homeorhesis conditions (A.4) and (A.5) hold.
Then the following assertions are valid.
(1) Function sχ (t) is such that h(Ψ (t, sχ ), s(t)) is independent of s(t), more specifically
h(Ψ (t, sχ ), s(t)) ≡ h(Ψ (t, sχ ), sχ (t)).
(A.6)
(2) Both ODE (A.1) and ODE
d[ν − Ψ (t, sχ )]/dt = H (ν, Ψ (t, sχ ), s(t))[ν − Ψ (t, sχ )]
(A.7)
where
∂h[Ψ (t, sχ ) + κ(ν − Ψ (t, sχ )), s(t)]
dκ
∂ν
0
have the unique solution Ψ (t, s) in function set Bn .
H (ν, Ψ (t, sχ ), s(t)) =
Z
1
(A.8)
Proof. Subtracting (A.3) from (A.2), one obtains d[Ψ (t, s) − Ψ (t, sχ )]/dt = h(Ψ (t, s), s(t)) − h(Ψ (t, sχ ), sχ (t))
or, equivalently,
d[Ψ (t, s) − Ψ (t, sχ )]/dt = [h(Ψ (t, s), s(t)) − h(Ψ (t, sχ ), s(t))]
+ [h(Ψ (t, sχ ), s(t)) − h(Ψ (t, sχ ), sχ (t))].
(A.9)
In view of (A.4) and (A.5), both the left-hand side of (A.9) and the term in the first bracket pair on the right-hand
side of (A.9) tend to zero when t → ∞. Since the term in the second bracket pair on the right-hand side of (A.9) is
independent of Ψ (t, s), it is identically equal to zero, i.e. (A.6) is valid. This proves assertion (1).
Application of (A.6) to (A.9) leads to
d[Ψ (t, s) − Ψ (t, sχ )]/dt = H [Ψ (t, s), Ψ (t, sχ ), s(t)][Ψ (t, s) − Ψ (t, sχ )]
(A.10)
where the matrix is described with expression (A.8) which stems from the well-known formula. Assertion (2) follows
from (A.10), Property A.1, and the meaning of functional set Bn (see the text below Remark A.2 for the latter). This
completes the proof. The limit relations (A.4) and (A.5) in the actual case when the steady-state approximation is not involved and the
system is living only in interval (t B , t D ) can be interpreted as follows. The relaxation times of functions associated
with both asymptotic representations (A.4) and (A.5) are much less than the life span t D − t B .
Remark A.5. According to Theorem A.1, if ODE (A.1) properly describes a living system, then the hypothesis and
all the assertions of the theorem must hold. This result and Property A.2 present a mathematical formulation of the
conditions necessary for the living-system ODE-based modelling. They show in particular what features function h
in (A.1) should have. More research is needed to further specify this function.
Remark A.5 summarizes the main result of the present appendix. In connection with this, we note the following
two issues.
Firstly, relation (A.6) that holds for every sufficiently small ks − sχ k can be read as the equality
∂h(Ψ (t, sχ ), w)/∂w ≡ 0,
at any sufficiently small kw − sχ k,
(A.11)
and, thus, as the insensitivity of the creode Ψ (t, sχ ) to the variation of values w of the exogenous signal (see (A.1)) in
a neighborhood of sχ (t). This fact contributes to the interpretation below.
Theorem A.1 reveals the specific meaning of the term “most favored” applied to the creode solution Ψ (t, sχ ) or the
corresponding exogenous signal sχ (cf., feature (iii)). Indeed, the theorem shows the following. Homeorhesis implies
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E. Mamontov et al. / Mathematical and Computer Modelling 44 (2006) 595–607
that a living system “chooses” the hypothetical value sχ of the actual exogenous-signal function s such that this value
determines the creode Ψ (t, sχ ) as the mode of the system which provides its insensitivity (A.6) (or (A.11)) to the
changes of s(t) locally, in a neighborhood of sχ (t). This insensitivity “protects” the living system from the influences
of the environmental variations s(t) − sχ (t). In other words, the above ODE vision suggests a specific formulation
(and the way to obtain it) for what is known as “the wisdom of the body” (cf., [26]).
Secondly, living systems develop not only in time but also in space. However, the above results can also be
interpreted in terms of the space-time evolution if ODE (A.1) is regarded in an appropriate function Banach space
rather than in Euclidean space. The key task at any of these settings is a derivation of function h in (A.1) which has the
proper mathematical properties (cf., Remark A.5) and is informally meaningful, i.e. fully relevant to the living system
underlying the model. More research on the route leading to this function is needed.
Even if one cannot specify function h with all of the aforementioned properties, one can apply assertion (2) of
Theorem A.1 as a practical recipe for modelling a living system, namely:
• the system is described with ODE (A.7) where matrix H (ν, Ψ (t, sχ ), s(t)) is not determined by means of (A.8)
but results from one or another approximation which is not related to function h used in (A.8);
• the creode trajectory Ψ (t, sχ ) is extracted from the corresponding experimental data;
• the actual trajectory Ψ (t, s) is determined as the unique solution of ODE (A.7) in the function set Bn (cf., assertion
(2) of the Theorem A.1).
The above approach, not fully consistent but reasonably pragmatic, is used in [20–22] to describe oncogenic
hyperplasia and the effect of the related therapy or drugs on the formation and disintegration of hyperplastic tumors.
The specific form of ODE (A.7) employed in this approach is
d[ν − νχ (t)]/dt = H [ν − νχ (t)]
(A.12)
where
H=
νT (t, νχ )[ν − νC (t, νχ )][ν − νT (t, νχ )]
1
,
η [νT (t, νχ ) − νχ (t)]{c(t)[νT (t, νχ ) − νC (t, νχ )]ν + νC (t, νχ )[νT (t, νχ ) − ν]}
(A.13)
ν is the concentration of the cells, νχ (t) = Ψ (t, sχ ) is the creode value of ν, 0 ≤ ν < νT (t, νχ ), νT (t, νχ ) is the
value of ν corresponding to the hyperplastic tumor, νC (t, νχ ) is the parameter of the cell population called the critical
concentration of the cells, νχ (t) ≤ νC (t, νχ ) < νT (t, νχ ), c(t) ∈ [1, 3) is the cell parameter which presents the extent
of the cell genotoxicity and which is coupled with νC (t, νχ ), and η > 0 is the lifetime of a cell at ν = νχ (t). Parameter
νC (t, νχ ) is such that νT (t, νχ ) − νC (t, νχ ) is proportional to |s(t) − sχ (t)|. The actual cell-population concentration
Ψ (t, s) is, according to assertion (2) of Theorem A.1, determined as the unique solution of ODE (A.12) in the function
set B1 . If the solution is known at one time point, say
Ψ (t, s)|t=t0 = ν0 ,
(A.14)
then Ψ (t, s) at any t 6= t0 can be evaluated from initial-value problem (A.12) and (A.14).
Importantly, when t → ∞, then Ψ (t, s) tends to either the creode νχ (t) or a tumor νT depending on whether
νχ (t) < Ψ (t, s) < νC (t, νχ ) or νC (t, νχ ) < Ψ (t, s) < νT (t, νχ ), respectively. Thus, the switching of νC (t, νχ )
between sufficiently high and sufficiently low values initiates the formation or disintegration of the tumor.
Also note that example (A.13) of matrix (A.8) allows one to interpret oncogeny as a genotoxically activated (i.e.
caused by νC (t, νχ ) or c(t)) homeorhetic dysfunction, in a complete agreement with the related results in biomedicine
(e.g., [27]). More generally, the form of the function (see (A.8)) in ODE (A.7) describes not only the creode but also
other stable states (i.e. homeorhetic dysfunctions) which can be approached or left by the living-system trajectory
Ψ (t, s) depending on the switching characteristics (such as the above critical concentration νC (t, νχ )). This important
option substantially extends the capabilities of evolution equation (A.1) in modelling various aspects of living systems.
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