Evolution of adaptation mechanisms: Adaptation energy, stress, and oscillating death

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arXiv:1512.03949v2 [q-bio.PE] 17 Mar 2016
Evolution of adaptation mechanisms: adaptation
energy, stress, and oscillating death
Alexander N. Gorban, Tatiana A. Tyukina
University of Leicester, Leicester, LE1 7RH, UK
Elena V. Smirnova, Lyudmila I. Pokidysheva
Siberian Federal University, Krasnoyarsk, 660041, Russia
Abstract
In 1938, H. Selye proposed the notion of adaptation energy and published “Ex-
perimental evidence supporting the conception of adaptation energy”. Adapta-
tion of an animal to different factors appears as the spending of one resource.
Adaptation energy is a hypothetical extensive quantity spent for adaptation.
This term causes much debate when one takes it literally, as a physical quan-
tity, i.e. a sort of energy. The controversial points of view impede the systematic
use of the notion of adaptation energy despite experimental evidence. Neverthe-
less, the response to many harmful factors often has general non-specific form
and we suggest that the mechanisms of physiological adaptation admit a very
general and nonspecific description.
We aim to demonstrate that Selye’s adaptation energy is the cornerstone of
the top-down approach to modelling of non-specific adaptation processes. We
analyse Selye’s axioms of adaptation energy together with Goldstone’s modifi-
cations and propose a series of models for interpretation of these axioms. Adap-
tation energy is considered as an internal coordinate on the ‘dominant path’ in
the model of adaptation. The phenomena of ‘oscillating death’ and ‘oscillating
remission’ are predicted on the base of the dynamical models of adaptation.
Natural selection plays a key role in the evolution of mechanisms of physio-
logical adaptation. We use the fitness optimization approach to study of the
distribution of resources for neutralization of harmful factors, during adaptation
to a multifactor environment, and analyse the optimal strategies for different
systems of factors.
Keywords: adaptation, general adaptation syndrome, evolution, physiology,
optimality, fitness
Email addresses: ag153@le.ac.uk (Alexander N. Gorban), tt51@leicester.ac.uk
(Tatiana A. Tyukina), seleval2008@yandex.ru (Elena V. Smirnova), pok50gm@gmail.com
(Lyudmila I. Pokidysheva)
Preprint submitted to Elsevier March 14, 2018

1. Introduction
Selye (1938a) introduced the notion of adaptation energy as the universal
currency for adaptation. He published “Experimental evidence supporting the
conception of adaptation energy” (Selye, 1938b): adaptation of an animal to dif-
ferent factors (sequentially) looks like spending of one resource, and the animal
dies when this resource is exhausted.
The term ‘adaptation energy’ contains an attractive metaphor: there is a
hypothetical extensive variable which is a resource spent for adaptation. At
the same time, this term causes much debate when one takes it literally, as a
physical quantity, i.e. as a sort of energy, and asks to demonstrate the physical
nature of this ‘energy’. Such discussions impede the systematic use of the notion
of adaptation energy even by some of Selye’s followers. For example, in the
modern “Encyclopedia of Stress” we read: “As for adaptation energy, Selye
was never able to measure it...” (McCarty & Pasak, 2000). Nevertheless, this
notion is proved to be useful in the analysis of adaptation (Breznitz, 1983;
Schkade & Schultz, 2003).
Without any doubt, adaptation energy is not a sort of physical energy. More-
over, Selye definitely measured the adaptation energy: the natural measure of it
is the intensity and length of the given stress from which adaptation can defend
the organism before adaptability is exhausted. According to Selye (1938b), “dur-
ing adaptation to a certain stimulus the resistance to other stimuli decreases”.
In particular, he demonstrated that “rats pretreated with a certain agent will
resist such doses of this agent which would be fatal for not pretreated controls.
At the same time, their resistance to toxic doses of agents other than the been
adapted decreases below the initial value.”
These findings were tentatively interpreted using the assumption that the
resistance of the organism to various damaging stimuli depends on its adaptabil-
ity. This adaptability depends upon adaptation energy of which the organism
possesses only a limited amount, so that if it is used for adaptation to a certain
stimuli, it will necessarily decrease.
Selye (1938b) concluded that “adaptation to any stimulus is always acquired
at a cost, namely, at the cost of adaptation energy.” No other definition of adap-
tation energy was given. This is just a resource of adaptability, which is spent
in all adaptation processes. The economical metaphors used by Selye, ‘cost’ and
‘spending’, were also seminal and their use was continued in many works. For
example, Goldstone (1952) considered adaptation energy as a “capital reserve
of adaptation” and death as “a bankruptcy in non-specific adaptation energy”.
The economical analogy is useful in physiology and ecology for analysis of
interaction of different factors. Gorban, Manchuk, & Petushkova (1987) anal-
ysed interaction of factors in human physiology and demonstrated that adap-
tation makes the limiting factors equally important. These results underly
the method of correlation adaptometry, that measures the level of adaptation
load on a system and allows us to estimate health in groups of healthy people
(Sedov et al., 1988). For plants, the economical metaphor was elaborated by
Bloom, Chapin, & Mooney (1985) and developed further by Chapin, Schulze, & H.A. Mooney
2

(1990). They also merged the optimality and the limiting approach and used the
notion of ‘exchange rate’ for factors and resources. For more details and connec-
tions to economical dynamics we refer to Gorban, Smirnova, & Tyukina (2010).
For systems of factors with different types of interaction (without limitation)
adaptation may lead to different results (Gorban et al., 2011). In particular, if
there is synergy between several harmful factors, then adaptation should make
the influence of different factors uneven and may completely exclude (compen-
sate) some of them.
In order to understand why we need the notion of adaptation energy in
modelling of physiology of adaptation, we have to discuss two basic approaches
to modelling, bottom-up and top-down.
•The bottom-up approach to modelling in physiology ties molecular and
cellular properties to the macroscopic behaviour of tissues and the whole
organism. Modern multiagent methods of modelling account for elemen-
tary interactions, and provide analysis how the rules of elementary events
affect the macroscopic dynamics. For example, Galle, Hoffmann, & Aust
(2009) demonstrate how the individual based models explain fundamen-
tal properties of the spatio-temporal organisation of various multi-cellular
systems. However, such models may be too rich and detailed, and typ-
ically, different model assumptions comply with known experimental re-
sults equally well. In order to develop reliable quantitative individual
based models, additional experimental studies are required for identify-
ing the details of the elementary events (Galle, Hoffmann, & Aust, 2009).
We suspect that for the consistent and methodical bottom-up modelling,
we will always need additional information for identification of the micro-
scopic details.
•Following the top-down approach, we start from very general integrative
properties of the whole system and then add some details from the lower
levels of organization, if necessary. It is much closer to the classical phys-
iological approach. A properly elaborated top-down approach creates the
background, the framework and the environment for the more detailed
models. We suggest, without exaggeration, that all detailed models need
the top-down background (like quantum mechanics, which cannot be un-
derstood without its classical limit). The top-down approach allows one
to relate the modelling process directly to experimental data, and to test
the model with clinical data (Hester et al., 2011). Therefore, the language
of the problem statement and the interpretation of the results is generated
using the top-down approach.
•To combine the advantages of the bottom-up and the top-down approaches,
the middle-out approach was proposed (Brenner, 1998; Kohl et al., 2010).
The main idea is to start not from the upper level but from the level which
is ready for formalization. That is the level where the main mechanisms
are known, and it is possible to develop an adequate mathematical model
without essential extension of experimental and theoretical basis. Then
3

we can move upward (to a more abstract integrative level) or downward
(to more elementary details), if necessary. Following Noble (2003) we sug-
gest that “reduction and integration are just two complementary sides of
the same grand project: to unravel and understand the ‘Logic of Life’.”
Selye (1938b) and later Goldstone (1952) used the notion of adaptation en-
ergy to represent the typical dynamics of adaptation. In that sense, they pre-
pared the theory of adaptation for mathematical modelling. The adaptation
energy is the most integrative characteristic for the models of top level. In this
work, we develop a hierarchy of top-down models following Selye’s findings and
further developments.
We follow Selye’s insight about adaptation energy and provide a ‘thermody-
namic-like’ theory of organism resilience that (just like classical thermodynam-
ics) allows for economic metaphors (cost, bankruptcy) and, more importantly,
is largely independent of a detailed mechanistic explanation of what is ‘going
on underneath’.
We avoid direct discussion of the question of whether the adaptation energy
is a ‘biological reality’, a ‘generalizing term’ for a set of some specific (unknown)
properties of an organism that provide its adaptation, or ‘just a metaphor’
similar to ‘phlogiston’ or ‘ether’, notions that were useful for description of
some phenomena but had no actual physical meaning as substances.
Moreover, we insist that the sense of the notion of adaptation energy is
completely described by its place in the system of models like the notion of mass
in Newtonian mechanics is defined by its place in the differential equations of
Newton’s laws. Selye did not write the equation of the adaptation energy but
his experiments and ‘axioms’ have been very ‘mathematical’. He proved that
(in some approximation) there is an extensive variable (adaptation resource)
which an organism spends for adaptation. This resource was measured by the
intensity and length of various stresses from which adaptation can defend the
organism.
2. ‘Axioms’ of adaptation energy
Selye, Goldstone and some other researchers formulated some of their discov-
eries and working hypotheses as ‘axioms’. These axioms, despite being different
from mathematical axioms, are used for fixing and securing sense. Selye’s ax-
ioms of Adaptation Energy (AE) (following Schkade & Schultz (2003)) are:
1. AE is a finite supply, presented at birth.
2. As a protective mechanism, there is some upper limit to the amount of
AE that an individual can use at any discrete moment in time. It can
be focused on one activity, or divided among other activities designed to
respond to multiple occupational challenges.
3. There is a threshold of AE activation that must be present to potentiate
an occupational response.
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Stressor
Stressor
Stressor
Stressor
Exhausted AE
Death
AE AE AE
Figure 1: Schematic representation of Selye’s axioms. The shield of adaptation spends AE for
protection from each stress. Finally, AE becomes exhausted, the animal cannot resist stress
and dies (The rat silhouette is taken from Wikimedia commons, File:Rat 2.svg.)
4. AE is active at two levels of awareness: a primary level at which creating
the response occurs at a high awareness level, with high usage of finite
supply of adaptation energy; and a secondary level at which the response
creation is being processing at a sub-awareness level, with a lower energy
expenditure.
Selye’s Axioms 1-3 are illustrated by Fig. 1.
Goldstone (1952) proposed the concept of a constant production or income
of AE which may be stored (up to a limit), as a capital reserve of adaptation.
He showed that this concept best explains the clinical and Selye’s own labora-
tory findings. According to Goldstone (1952), it is possible that, had Selye’s
experimental animals been asked to spend adaptation at a lesser rate (below
their energy income), they might have been able to cope successfully with their
stressor indefinitely. The whole systems of adaptation reactions to weaker fac-
tors was systematized by Garkavi, Kvakina, & Ukolova (1979). On the basis of
this system, Garkavi et al. (1998) developed the activation therapy, which was
applied in clinic, aerospace and sport medicine.
Goldstone’s findings may be formulated as a modification of Selye’s axiom 1.
Their difference from Selye’s axiom 1 is illustrated by Fig. 2 (compare to Fig. 1).
We call this modification Goldstone’s axiom 1’:
•AE can be created, though the income of this energy is slower in old age;
•It can also be stored as Adaptation Capital, though the storage capacity
has a fixed limit.
•If an individual spends his AE faster than he creates it, he will have to
draw on his capital reserve;
•When this is exhausted he dies.
5

Stressor
Stressor
AE AE AE
Figure 2: Schematic representation of Goldstone’s modification of Selye’s axioms: AE can be
recovered and adaptation shield may persist if there is enough time and reserve for recovery.
3. Factor-resource basic model of adaptation
Let us start from a simple (perhaps, the simplest) model with two phase
variables, the available free resource (AE) r0and the resource supplied for the
stressor neutralization, r. There are also four processes: degradation of the
available resource, degradation of the supplied resource, supply of the resource
from the storage r0to the allocated resource r, and production of the resource
for further storage (r0). The equations are:
dr
dt=−kdr+kr0(f−r)h(f−r);
dr0
dt=−kd0r0−kr0(f−r)h(f−r) + kpr (R0−r0),
(1)
where
•kdris the rate of degradation of resource supplied for the stressor neutral-
ization, where kdis the corresponding rate constant;
•kd0r0is the rate of degradation of the stored resource, where kd0is the
corresponding rate constant, we assume that kd≥kd0;
•kr0(f−r)h(f−r) is the rate of resource supply for the stressor neutral-
ization, where kis the supply constant;
•h(f−r) is the Heaviside step function;
•kpr(R0−r0) is the resource production rate, where kpr is the production
rate constant.
Let us notice that:
•if r0≥R0then dr0/dt≤0,
6

•if r0= 0 then dr0/dt≥0,
•if r= 0, r0≥0, then dr/dt≥0,
•if r=fthen dr/dt≤0.
Therefore, the rectangle Dgiven by inequalities 0 ≤r≤f, 0 ≤r0≤R0is posi-
tively invariant with respect to system (1): if the initial values (r(t0), r0(t0)) ∈D
for some time moment t0then the solution (r(t), r0(t)) ∈Dfor t > t0.
For large fthere exist a stable steady state in Dwith
r0≈kprR0
kf ;r≈kr0f
kd
≈kprR0
kd
.
AE is never exhausted even when f→ ∞. Immortality at infinite load is
possible. Something is wrong in the model. AE production should decrease
for large non-compensated stressors ψ=f−r. Let us modify the production
term in (1) and add a fitness (well-being) W. This fitness (well-being), is equal
to one when the stressor load is compensated and goes to zero when the non-
compensated value of the stressor load ψ=f−rbecomes sufficiently large. Let
us choose the following form of Wfor one-factor model:
W(ψ) = 1−ψ
ψ0,0≤ψ≤ψ0.(2)
Fitness W(ψ) is a linear function on the interval 0 ≤ψ≤ψ0. It takes its
maximal value 1 at point ψ= 0 (completely compensated stressors) and vanishes
at ψ=ψ0(Fig. 3).
Formally, it may be continued to the whole line by constants: W= 1 for
ψ < 0 and W= 0 for ψ > ψ0:
W(ψ) = 1−ψh(ψ)
ψ0h1−ψh(ψ)
ψ0.
Nevertheless, it is convenient to use the simplest linear function (2) and
analyse the system at the borders ψ= 0 and ψ= 1 separately.
The modified system of equations has the form:
dr
dt=−kdr+kr0(f−r)h(f−r);
dr0
dt=−kd0r0−kr0(f−r)h(f−r) + kpr (R0−r0)W(f−r),
(3)
where the fitness function W(ψ) is given by (2).
4. Problems in definition of instant individual fitness
We use an individual’s fitness Wto measure the wellbeing (or performance)
of an organism. Moreover, this is an instant value, defined for every time mo-
ment. Defining of the instant measure of an individual’s performance is a highly
7

ψ=f-r
W(ψ)
߰଴
ͳ
Death
(W<0)
Figure 3: The fitness function for system (3). ψ0is the critical value of stressor’s intensity. If
f≤ψ0then life is possible without adaptation: for zero AE supply Wremains positive.
non-trivial task. The term ‘fitness’ is widely used in mathematical biology in
essentially another sense based on the averaging of reproduction rate over a
long time (Haldane, 1932; Maynard-Smith, 1982; Metz, Nisbet, & Geritz, 1992;
Gorban, 2007). This is Darwinian fitness. It is non-local in time because it
is the average reproduction coefficient in a series of generations and does not
characterize an instant state of an individual organism.
The synthetic evolutionary approach starts with the analysis of genetic vari-
ation and studies the phenotypic effects of that variation on physiology. Then it
goes to the performance of organisms in the sequence of generations (with ad-
equate analysis of the environment) and, finally, it has to return to Darwinian
fitness Lewontin (1974). The physiological ecologists are focused, first of all, on
the observation of variation in individual performance (Pough, 1989). In this
approach we have to measure the individual performance and then link it to the
Darwinian fitness.
The connection between individual performance and Darwinian fitness is
not obvious. Moreover, the dependence between them is not necessarily mono-
tone. This observation was formalized in the theory of r−and K−selection
(MacArthur & Wilson, 1967; Pianka, 1970). The terminology refers to the equa-
tion of logistic growth: ˙
N=rN (1 −N
K) (Kis the ‘carrying capacity’ and rthe
maximal intrinsic rate of natural increase). Roughly speaking, Kmeasures the
competitive abilities of individuals, and rmeasures their fecundity. Assuming
negative correlations between rand K, we get a question: what is better in
the Darwinian sense: to increase individual competitive abilities or to increase
fecundity? Earlier, Fisher (1930) formulated a particular case of this problem
as follows: “It would be instructive to know not only by what physiological
mechanism a just apportionment is made between the nutriment devoted to the
gonads and that devoted to the rest of the parental organism, but also what
circumstances in the life-history and environment would render profitable the
diversion of a greater or lesser share of the available resources towards repro-
duction.” The optimal balance between individual performance and fecundity
depends on environment. Thus, Dobzhansky (1950) stated that in the trop-
ical zones selection typically favors lower fecundity and slower development,
8

whereas in the temperate zones high fecundity and rapid development could
increase Darwinian fitness.
Nevertheless, the idea that the states of an organism could be linearly or-
dered from bad to good performance (wellbeing) is popular and useful in applied
physiology. The coordinate on this scale is also called ‘fitness’. Several indica-
tors are measured for fitness assessment and then the fitness is defined as a
composite of many attributes and competencies. For example, for fitness as-
sessment in sport physiology these competencies include physical, physiological
and psychomotor factors (Reilly & Doran, 2003). The balance between various
components of sport-related instant individual fitness depends upon the specific
sport, age, gender, individual history and even on the role of the player in the
team (for example, for football).
Similarly, the notion ‘performance’ in ecological physiology is ‘task–depen-
dent’ (Wainwright, 1994) and refers to an organism’s ability to carry out specific
behaviors and tasks (e.g., capture prey, escape predation, obtain mates). Di-
rect instant measurement of Darwinian fitness is impossible but it is possible to
measure various instant performances several times and treat them as the com-
ponents of fitness in the chain of generations. Arnold (1983) proposed several
criteria for selection of the good measure of performance in the evolutionary
study: (1) the measure should be ecologically relevant, i.e. it measures success
in the ecologically important behavior significant for survival and reproductive
output; (2) the measure should be phylogenetically interesting, i.e. it captures
the differences between taxa and the difference between higher taxa is larger
than for closed taxa, at least, for some types of performance. The relations
between performance and lifetime fitness are sketched on flow-chart (Fig. 4)
following Wainwright (1994) with minor changes. Darwinian fitness may be
defined as the lifetime fitness averaged in a sequence of generations.
The idea of individual fitness is intensively used in conservation physiology
(Wikelski & Cooke, 2006). An important problem is to determine how sin-
gle intensive periods of stress influence individual fitness. Wikelski & Cooke
(2006) stressed that when the link between baseline physiological traits and
fitness is known, conservation managers can use physiological traits as indi-
cators to predict and anticipate future problems. Ecological success is cou-
pled to environmental conditions via the sensitivity of physiological systems
(Seebacher & Franklin, 2012). Ideally, individual fitness is maximized when the
organism can perform at a constant and optimal level despite environmental
variability, but this is impossible in the changing world for several reasons: (i)
adaptation requires time and there is a lag between the changes in environment
and the adaptive response, (ii) adaptation has a cost and excessive adaptation
load may decrease performance because of this cost, and (iii) adaptation has its
limits and even in the most plastic organisms, the capacity to compensate for
environmental change is bounded.
We use the instant individual fitness (wellbeing) Was a characteristic of the
current state of the organism, reflecting the non-optimality of its performance:
W= 1 means the maximal achievable performance, W= 0 means inviability
(death). If the organism lives at some level of Wthen we can consider Was a
9

Genotype
Phenotype
Performances
Resource use
Survival
Reproductive
output
Lifetime
fitness
Environment
Ecological challenges
Ecological environment
Figure 4: Flow diagram showing the paths through from genotype to Darwinian fitness.
Genotype in combination with environment determines the organismal design (the pheno-
type) up to some individual variations. Phenotype determines the limits of an individual’s
ability to perform day-to-day behavioral answer to main ecological challenges (performances).
Performance capacity interacts with the given ecological environment and determines the re-
source use, which is the key internal factor determining r−and K−components of fitness,
reproductive output and survival.
factor in the lifetime fitness. Such a factorization assumes that the physiological
state of the organism acts independently of other factors to determine fitness.
This assumption follows the ideas of Fisher (1930). The basic assumptions of
Fisher’s model were analysed by Haldane (1932). ‘Independence’ here is con-
sidered as multiplicativity, like in probability theory. Of course, the hypothesis
of independence is never absolutely correct, but it gives a good initial approx-
imation in many areas, from data mining (na¨ıve Bayes models) to statistical
physics (non-correlated states).
This is the qualitative explanation of the instant individual fitness W. It is
the most local in time level in the multiscale hierarchy of measures of fitness:
instant individual fitness to individual life fitness to Darwinian fitness in the
chain of generations. The proper language for discussion of the individual fitness
gives the idea of particular performances, these are abilities of the organism to
answer various specific ecological challenges. The instant individual fitness aims
to combine various indicators of different performances into one quantity.
The quantitative definition of the Wscale is given by its place in the equa-
tions. The change of the basic equation will cause the change of the quantitative
definition. Now, we are far from the final definition of W. Moreover, it is plau-
sible that for different purposes we may need different definitions of W.
5. Dangerous borders
The fitness takes the maximal value W= 1 if the factor is fully compensated,
f=r. Due to equations (3) if f=rand r≥0 then dr/dt=−kdr≤0 and
10

a)
r0
f-ψ0
݀ݎ ݀ݐ
Τ
ൌ Ͳ

ݎ଴ൌ ݇ௗݎ
݇ሺ݂ െ ݎሻ
r0=R0
r
f
Dangerous
border (death)
݀ݎ ݀ݐ
Τ
൏ Ͳ
R0
r=f
W=0
Save border
(survival)
݀ݎ ݀ݐ
Τ
൐ Ͳ
W<0 W>0
b)
r0
f-ψ0≤0
Τ
݀ݎ ݀ݐ ൌ Ͳ
r0=R0
r
f
R0
r=f
W=0
Save border
(survival)
Τ
݀ݎ ݀ݐ ൐ Ͳ
Τ
݀ݎ଴݀ݐ ൏ Ͳ
Τ
݀ݎ ݀ݐ ൏ Ͳ
Τ
݀ݎ଴݀ݐ ൐ Ͳ
Τ
݀ݎ଴݀ݐ ൌ Ͳ S
c)
r0
݀ݎ ݀ݐ
Τ
ൌ Ͳ
݀ݎ଴݀ݐ
Τ
ൌ Ͳ
r0=R0
S
U
r
f
Life area
(positively
invariant)
Dangerous
border (death)
Separatrix
R0
r=f
W=0
Save border
(survival)
f-ψ0
d)
r0
݀ݎ ݀ݐ
Τ
ൌ Ͳ
݀ݎ଴݀ݐ
Τ
ൌ Ͳ
r0=R0
r
f
All trajectories
lead to death
R0
r=f
W=0
Save
border
f-ψ0
Death
Figure 5: Safe and dangerous borders for adaptation system (3) for q > 0. The r-nullcline
cuts the border of death W= 0 (r=f−ψ0) into two parts: ˙
W < 0 (dangerous border, red)
and ˙
W > 0 (safe border, green) (a). The nullclines have in this case (a) unique intersection
point Sin D(that is the stable equilibrium). If f < ψ0then the whole border is safe (b).
If the r- and r0-nullclines have two intersections, the stable (S) and unstable (U) equilibria
(c), then the separatrix of the unstable equilibrium Useparates the area of attraction of the
dangerous border (area of death) from the area of attraction of stable equilibrium (life area)
(c). If there exists no intersection of the nullclines in the rectangle (d) then all the trajectories
are attracting to the dangerous border.
dWdt≤0. Therefore, the fitness Wcannot exceed the value 1 if it is initially
below 1.
The line W= 0 (i.e. f−r=ψ0) is a border of death. If Wbecomes negative,
it means death. On this border,
If r0< kd
f−ψ0
ψ0
then dr
dt<0 and dW
dt<0;
If r0> kd
f−ψ0
ψ0
then dr
dt>0 and dW
dt>0;
The situation when W= 0 and dW/dt < 0 leads to death. Therefore,
this part of the border (r0< kd(f−ψ0)/ψ0) is called the dangerous border.
On the contrary, if W= 0 but dW/dt > 0 it means survival and this border
(r0> kd(f−ψ0)/ψ0) is safe. The intersection point of the border of death
and the r-nullcline of system (3) separates the safe part of the border from the
dangerous part (Fig. 5a).
If f≤ψ0then the whole border of death belongs to the half-plane r≤0
(Fig 5b). In this case, all the borders of the rectangle D(0 ≤r≤f, 0 ≤r0≤
R0) are repulsive and the motion remains in Dforever, if it starts in D. Below
11

we consider the case 0 < ψ0< f . Let us analyse the system (3) in the rectangle
Qgiven by the inequalities
Q: 0 ≤r, f −ψ0≤r≤f , 0≤r0≤R0.(4)
In the rectangle Qthe Heaviside functions in system (3) could be deleted and
this system takes a simple bilinear form
dr
dt=−kdr+kr0(f−r);
dr0
dt=−kd0r0−kr0(f−r) + kpr (R0−r0)1−f−r
ψ0,
(5)
Qis not necessarily positively invariant with respect to (5). The system may
leave Qthrough the dangerous border.
The nullclines of this system (5) in Qare plots of monotonic functions r0(r).
The r-nullcline is, for r < f , monotonically growing convex function of r:
−kdr+kr0(f−r) = 0,or r0=kdr
k(f−r)=kd
kf
f−r−1.
The r0-nullcline is
−kd0r0−kr0(f−r) + kpr (R0−r0)1−f−r
ψ0= 0,or
r0=kprR0
qψ0 1−
1
q(kd0+kψ0)
r−(f−ψ0) + 1
q(kd0+kψ0)!,
where q=1
ψ0kpr −k6= 0.
The product qψ0=kpr −kψ0is the difference between the adaptation energy
production rate constant kpr and the supply coefficient kψ0at the critical value
f−r=ψ0(the supply rate is k(f−r)r0).
If q= 0 then the r0-nullcline is a straight line
r0=kprR0
ψ0
r−(f−ψ0)
kd0+kψ0
.
Geometry of the phase portraits is schematically presented in Fig. 5b,c,d.
The nullclines are monotonic, the r-nullcline is convex, and for the case q > 0 the
r0-nullcline is concave. The area between the nullclines is positively invariant.
The phase portrait transforms from Fig. 5b to Fig. 5c and d when the pressure
of factor fincreases starting from safe values f≤ψ0to high values f≫ψ0.
6. Resource and reserve
Selye, Goldstone and other researchers stressed that there are different levels
of the adaptation energy supply, with lower and higher energy expenditure.
12

ݎ଴
ܤ௢Ȁ௖
ܤ௢Ȁ௖=1
ܤ௢Ȁ௖=0
ݎ ݎ
ҧ
Figure 6: Resource – reserve hysteresis. Hysteresis of reserve supply: if Bo/c = 0 then reserve
is closed and if Bo/c = 0 then reserve is open. When r0decreases and approaches rthen the
supply or reserve opens (if it was closed). When r0<rincreases and approaches rthen the
supply of reserve closes (if it was open).
Garkavi, Kvakina, & Ukolova (1979) insisted that there are many levels at lower
intensity of stressors, and created the ‘periodic table’ of the adaptation reactions.
Nevertheless, we propose to formalize, first, the two-state hypothesis.
There are two storages of AE: resource (which is always available if it is not
empty) and reserve (which becomes available when the resource becomes too
low). The Boolean variable Bo/c describes the state of the reserve storage: if
Bo/c = 0 then the reserve storage is closed and if Bo/c = 1 then the reserve
storage is open. There are two switch lines on the phase plane (r, r0): r0=r
(the lower switch line that serves to opening the reserve storage) and r0=r
(the upper switch line that serves to closing the reserve storage). When the
available resource r0decreases and approaches rfrom above then the supply or
reserve opens (if it was closed). When the available resource r0< r increases
and approaches rfrom below then the supply of reserve closes (if it was open).
For r0< r the reserve is always open, Bo/c = 1 and for r0> r the reserve
is always closed, Bo/c = 0 (Fig. 6). These rules together with the following
equations describe the system in the rectangle Q(4).
dr
dt=−kdr+kr0(f−r);
dr0
dt=−kd0r0−kr0(f−r) + krv Bo/c rrv (R0−r0) + kpr (R0−r0)W;
drrv
dt=−kd1rrv −krv Bo/c rrv (R0−r0) + kpr1(Rrv −rr v)W,
(6)
where (r, r0) belongs to the rectangle Q(4), 0 ≤rrv ≤Rrv is the amount of
reserve, Rrv =const is the upper limit of the reserve and W= 1 −f−r
ψ0if we
accept the particular simple form of fitness function (2).
For dynamics of r0, the additional supply of AE from the reserve looks like
the increase of the well-being Wby krvrrv /kpr : after joining the last two terms
13

Figure 7: Oscillating recovery (a,c) and oscillating death (b,d) near the border W= 0 for
the systems with large reserve. (Horizontally stretched sketch.) In case (a) both r, r > r∗, in
case (b) both r, r < r∗, and in cases (c),( d) r > r∗> r, where r∗is the value of r0, which
separates the safe border from the dangerous border on the line W= 0 (8). The straight
angles of possible velocities are presented for motions without research supply in cases (a) and
(b)
in the second equation of (6) we get
dr0
dt=−kd0r0−kr0(f−r) + kpr (R0−r0)W+krv Bo/crrv
kpr .(7)
Let us analyse the impact of reserve on the dynamics of adaptation in the
small vicinity of the border of death W= 0. For simplicity, consider the case
with sufficiently large reserve and fast reserve recovery.
There are three qualitatively different cases of the motion in the interval
r≥r0≥rnear the border W= 0:
•r, r > r∗and the motion goes above both nulclines (Fig. 7a);
•r, r < r∗and the motion goes below the r-nulcline but above the r0-
nulcline (Fig. 7b);
•r > r∗> r and the motion intersects r-nulcline (Fig. 7c,d).
Here, r∗is the value of r0, which separates the safe border from the dangerous
border on the line W= 0,
r∗=kd
k
f−ψ0
ψ0
.(8)
In all these cases the motion oscillates between the lines r0=rand r0=r
(Fig. 7a). When the motion with closed reserve supply (Bo/c = 0) reaches the
line r0=rthen the reserve supply switches on (Bo/c = 1), the value of r0goes
up fast and quickly achieves r(because of the assumption of large reserve). The
14

value of rdoes not change significantly during this ‘jump’ of r0from rto r.
When the motion with open reserve supply (Bo/c = 1) reaches the line r0=r
then supply of reserve switches off (Bo/c = 0) and the value of r0decreases.
(Note, that if the motion is sufficiently close to the border W= 0 then it is
above the nullcline of r0on the plane (r, r0), Figs. 5, 7.)
Consider the motion which starts on the line r0=rwith open reserve supply.
The motion returns to the same line r0=rafter the cycle: ‘jump up’ to the
line r0=r, switch reserve supply off and ‘move down’ without reserve supply to
the line r0=r, but the value of rmay change. If this change ∆r > 0 then the
system moves from the border W= 0 (oscillating recovery, Fig. 7a,c). If ∆r < 0
then the system moves to the border W= 0 (oscillating death, Fig. 7b,d).
If we combine the cases Fig. 7c (close to the border W= 0) and d (at some
distance from this border) then we can find the stable closed orbit for some
combination of parameters in the limit of large reserve and fast reserve recovery.
Such an orbit is presented in Fig. 8a (numerical calculation). If we decrease the
reserve recovering constant kpr1(and do not change other constants) then the
closed orbit may become larger with longer time of reserve supply (Fig. 8b).
The further decrease of kpr1leads to destruction of the closed orbit and the
oscillating death appears (Fig. 8d). The values of parameters were chosen just
for numerical example.
Fig. 8c demonstrates an important effect: the tra jectories spend a long time
near the places where cycles appear for different values of constants (see Fig. 8a
and Fig. 8b) and go to the attractor (here it is death) after this delay. The
delayed relaxation is a manifestation of the so–called ‘critical retardation’: near
a bifurcation with the appearance of new ω-limit points, the trajectories spend
a long time close to these points (Gorban, 2004).
The models based on Selye’s idea of adaptation energy demonstrate that the
oscillating remission and oscillating death do not need exogenous reasons. These
phenomena have been observed in clinic for a long time and now attract atten-
tion in mathematical medicine and biology. For example, Zhang, Wahl, & Yu
(2014) demonstrated recently, on a more detailed model of adaptation in the im-
mune system, that cycles of relapse and remission, typical for many autoimmune
diseases, arise naturally from the dynamical behavior of the system. The notion
of ‘oscillating remission’ is used also in psychiatry (Gudayol-Ferr´e, Gu`ardia-Olmos, & Per´o-Cebollero,
2015).
7. Distribution of adaptation energy in multifactor systems
Usually, organisms experience a load of many factors, where the effect of
one factor could depend on the loads of all other factors. We define a harm-
ful factor or ‘stressor’ as a noxious stimulus and the ‘stress response’ of an
organism as a suite of physiological and behavioral mechanisms to cope with
stress (Wikelski & Cooke, 2006). Revealing and description of important fac-
tors may be a non-trivial task because any biological pattern is correlated with
a large number of abiotic and biotic patterns. Some of them are known, though
15

Figure 8: Oscillations near the border of death for system (6) in pro jection onto the (r, r0)
plane (the reserve coordinate rrv is hidden). For each case (a), (b), and (c) several trajectories
are plotted together (central plots) and separately (side plots). At the initial points of all
trajectories the reserve is full, rrv =Rr v. For all cases r= 2, r= 0.5, R0= 10, Rrv = 5,
kd= 1, kd0= 0.1, kd1= 0.1, k= 0.5, kpr = 2, krv = 2, ψ0= 7, and f= 10. For case (a)
kpr1= 18 (stable oscillation), for case (b) kpr 1= 7 (stable oscillations with longer orbit), for
case (c) the closed orbit vanishes and the trajectories cross the borders of death (kpr1= 3.6).
16

many are unknown. Correlations are not sufficient for extraction of main fac-
tors and the special effort and experimental study are needed to reveal causality
(Seebacher & Franklin, 2012).
The effect of action of several factors may be far from additive. There are
various mechanisms of interaction between factors in their action. The discovery
of the first non-additive interaction between factors was done by Carl Spren-
gel in 1828 and Justus von Liebig in 1840 (van der Ploeg, B¨ohm, & Kirkham,
1999). They proposed ‘the law of the minimum’ (known also as ‘Liebig’s law’).
This law states that growth is controlled by the scarcest resource (limiting fac-
tor) (Salisbury, 1992). It is widely known that not all systems of factors satisfy
the law of the minimum. For example, some harmful factors can intensify ef-
fects of each other (effect of synergy means that the harm is superadditive).
The colimitation effects are also widely known (Wutzler & Reichstein, 2008).
Gorban et al. (2011) analysed and compared adaptation to Liebig’s and syner-
gistic systems of factors. They formalized the idea of synergy for multifactor
systems, introduced generalized Liebig’s systems and studied distribution of AE
for neutralization of the load of many factors. For this purpose, the optimality
principle was used. Tilman (1980) studied resource competition. He developed
an equilibrium theory based on classification of interaction in pairs of resources.
According to Tilman (1980) they may be: (1) essential, (2) hemi-essential, (3)
complementary, (4) perfectly substitutable, (5) antagonistic, or (6) switching.
He also used the idea of optimality.
Evolutionary approach aims to give a universal key to the problem of opti-
mality in biology (Haldane, 1932; Maynard-Smith, 1982; Gorban & Khlebopros,
1988). The universal measure of optimality is Darwinian fitness, that is the re-
production coefficient averaged in a long time (Gorban, 2007) with some analytic
simplifications, when it is possible (Karev & Kareva, 2014), and with known
generalizations for vector distributions (Gorban, 1984; Metz, Nisbet, & Geritz,
1992). However, there is no universal rule to measure various traits of organ-
isms by the changes in the average reproduction coefficient, despite exerted
efforts, development of special methods, and gaining some success (Haldane,
1954; Waxman & Welch, 2005; Kingsolver & Pfennig, 2007; Shaw et al., 2008;
Karev & Kareva, 2014). There may be additional difficulties because the evo-
lutionary optimality is not necessarily related to organisms, and the non-trivial
question arises: “what is optimal?” Another difficulty is caused by possible
non-stationarity of the optimum: selected organisms change their environment
and become non-optimal on the background of the new ecological situation
(Gorban, 1984). Nevertheless, the idea of fitness is proved to be very useful.
Fitness functions are defined for different situations as intermediates between
the (observable) traits of the animal and the average reproduction coefficient.
The factors-resource models with the fitness optimization allow us to trans-
late the elegant dynamic approach of the mathematical theory of evolution into
physiological language. The key idea is to use statistical properties of physio-
logical data instead of the data themselves. Correlations and variances are often
more reliable characteristics of stress and adaptation than the values of physio-
logical indicators (Gorban, Manchuk, & Petushkova, 1987; Gorban, Smirnova, & Tyukina,
17

2010; Gorban et al., 2011; Censi et al., 2011; Bernardini et al., 2013).
For formal definitions of Liebig’s and synergistic systems of factors the notion
of individual and instant fitness is used. We consider organisms that are under
the influence of several factors Fiwith intensities fi(i= 1, ...q). For definiteness,
assume that all the factors are harmful (this is just the sign convention plus
monotonicity assumption). AE supplied for neutralization of ith factor is ri
and fitness Wis a smooth function of qvariables ψi=fi−ri≥0. This means
that the factors are measured in the general scale of AE units. Comparability of
stressors of different nature was empirically demonstrated and studied by Selye
(1938b). It was a strong argument for introduction of AE. The value fi−ri=
0 is optimal (the fully compensated factor), and any further compensation is
impossible.
Assume that the vector of variables (ψ1,...,ψq) belongs to a convex subset
Uof the positive orthant Rq
+, and Wis defined in U. Harmfulness of all factors
means that
∂W (ψ1,...,ψq)
∂ψi
<0 for all i= 1,...,q and (ψ1,...,ψq)∈U.
Definition 1. A system of factors is Liebig’s system, if there exists a function
of one variable w(ψ)such that
W(ψ1,...,ψq) = wmax
1≤i≤q{fi−airi}.(9)
A system of factors is anti-Liebig’s system, if there exists a function of one
variable w(ψ)such that
W(ψ1,...,ψq) = wmin
1≤i≤q{fi−airi}.(10)
In Liebig’s systems fitness depends on the worst factor pressure. In anti-Liebig’s
systems fitness depends on the easiest factor pressure and the factors affect the
organism only together, in strong synergy.
To generalize these polar cases of Liebig’s and anti-Liebig’s system, recall the
notions of quasiconvex and quasiconcave functions. A function Fon a convex
set Uis quasiconvex (Greenberg & Pierskalla, 1971) if all its sublevel sets are
convex. It means that for every X, Y ∈U
F(λX + (1 −λ)Y)≤max{F(X), F (Y)}for all λ∈[0,1] (11)
In particular, a function Fon a segment is quasiconvex if all its sublevel sets
are segments.
A function Fon a convex set Uis quasiconcave if −Fis quasiconvex. Direct
definition is as follows: A function Fon a convex set Uis quasiconcave all its
superlevel sets are convex. It means that for every X, Y ∈U
F(λX + (1 −λ)Y)≥min{F(X), F (Y)}for all λ∈[0,1] (12)
18

In particular, a function Fon a segment is quasiconcave if all its superlevel sets
are segments.
For Liebig’s system the superlevel sets of Ware convex, therefore, W(ψ1,...,ψq)
is quasiconcave.
For anti-Liebig’s system the sublevel sets of Ware convex, therefore, W(ψ1,...,ψq)
is quasiconvex.
Definition 2. A system of factors is generalized Liebig’s system if W(ψ1,...,ψq)
is a quasiconcave function.
A system of factors is a synergistic one, if W(ψ1,...,ψq)is a quasiconvex
function.
Proposition 1. A system of factors is generalized Liebig’s system, if and only
if for any two different vectors of factor pressures ψ= (ψ1, ...ψq)and φ=
(φ1, ...φq)(ψ6=φ) the value of fitness at the average point (ψ+φ)/2is greater,
than at the worst of points ψ,φ:
Wψ+φ
2>min{W(ψ), W (φ)}.(13)
Proposition 2. A system of factors is a synergistic one, if for any two different
vectors of factor pressures ψ= (ψ1, ...ψq)and φ= (φ1, ...φq)(ψ6=φ) the value
of fitness at the average point (ψ+φ)/2is less, than at the best of points ψ,φ:
Wψ+φ
2<max{W(ψ), W (φ)}.(14)
Distribution of the supplied AE between factors should maximize the fitness
function Wwhich depends on the compensated values of factors, ψi=fi−ri.
The total amount rof the allocated AE is given:
W(f1−r1, f2−r2, ...fq−rq)→max ;
ri≥0, fi−ri≥0, Pq
i=1 ri≤r . (15)
Analysis of this optimization problem (Gorban, Manchuk, & Petushkova,
1987; Gorban, Smirnova, & Tyukina, 2010) leads to the following statements
(Gorban et al., 2011) which sound paradoxical (if law of the minimum is true
then the adaptation makes it wrong; if law of the minimum is significantly
violated then the adaptation decreases these violations):
•Law of the minimum paradox: If for a randomly selected pair, (‘State of
environment – State of organism’), the law of the minimum is valid (every-
thing is limited by the factor with the worst value) then, after adaptation,
many factors (the maximally possible amount of them) are equally impor-
tant.
•Law of the minimum inverse paradox: If for a randomly selected pair,
(“State of environment – State of organism”), many factors are equally
19

} }
} }
Equalizing of factors in optimum
Imbalance of factors in optimum
Figure 9: Distribution of AE for neutralization of several harmful factors for different types
of interactions between factors: (a) Liebig’s system (the fitness Wdepends monotonically on
the maximal non-compensated factor load only), (b) generalized Liebig’s system (the fitness
Wis a quasiconcave function of non-compensated factors loads), (c) anti-Liebig system (the
fitness Wdepends monotonically on the minimal non-compensated factor load only), and
(d) synergistic system (the fitness Wis a quasiconcave function of non-compensated factors
loads). Interval Lrepresents the area of optimization. ‘Harmful’ means that ∂W/∂fi<0 for
all factors.
important and superlinearly amplify each other then, after adaptation, a
smaller amount of factors is important (everything is limited by the factors
with the worst non-compensated values, the system approaches the law of
the minimum).
These properties of adaptation are illustrated by Fig. 9.
Adaptation of an organism to Liebig’s system transforms the one-dimensional
picture with one limiting factor into a high dimensional picture with many
important factors. Therefore, the well-adapted Liebig’s systems should have
less correlations between their attributes than in stress. The variance (fluc-
tuations) increases in stress. The large collection of data which supports this
property of adaptation in Liebig’s system was collected since the first publication
(Gorban, Manchuk, & Petushkova, 1987) and was reviewed by Gorban, Smirnova, & Tyukina
(2010).
Let us mention several new findings. Censi et al. (2011) proposed using the
connectivity of correlation graphs in gene regulation networks as an indicator
20

of analysis of illnesses and demonstrated the validity of this approach on pa-
tients with atrial fibrillation. Bernardini et al. (2013) studied mitochondrial
network genes in the skeletal muscle of amyotrophic lateral sclerosis patients
and found correlations of gene activities for ill patients higher than in control.
Kareva, Morin, & Castillo-Chavez (2015) found signs of this general effect in
their study of consumer–resource type models and analysis of population man-
agement strategies and their efficacy with respect to population composition.
Bezuidenhout, van Antwerpen, & Berry (2012) used this effect to measure the
health of soil and validated this approach. Pareto correlation graphs, includ-
ing only the highest 20% of correlation coefficients, were particularly useful
in depicting the larger aggregated manageability and measurability of soils.
Pokidysheva & Ignatova (2013) used analysis of dimension of the data cloud in
evaluation of human immune systems for patients with allergic disease, either
complicated or not complicated by clamidiosis. The patterns of population fluc-
tuations are considered as leading indicators of catastrophic shifts and extinction
in deteriorating environments (Dakos et al., 2010; Drake & Griffen, 2010). The
integration level in the redox in a tissue was systematically studied (Costantini,
2014).
Chen et al. (2012) analysed microarray data of three diseases and demon-
strated that when the system reached the pre-disease state then:
1. There exists a group of molecules, i.e., genes or proteins, whose average
correlation coefficients of molecules drastically increase in absolute value.
2. The average standard deviations of molecules in this group drastically
increase.
3. The average correlation coefficients of molecules between this group and
any others drastically decrease in absolute value.
The observation 1 (increase of the correlations in the dominant group) and 2
(increase of the variance in the dominant group) is in agreement with many
of our previous results for different systems and with results of Censi et al.
(2011), whereas the interesting observation 3 (decrease of the correlations be-
tween the dominant group and others, i.e. isolation of the dominant group)
seems to be less universal (see, for example, the correlation graphs published by
Gorban, Smirnova, & Tyukina (2010)).
Rybnikova & Rybnikov (2012) applied the method of measurement of stress
based on the Liebig’s paradox to assessing of societal stress in Ukraine. They
diagnosed significant stress and dysadaptation increase before the obvious crit-
ical events occur (the report was published in 2012, a year before crisis). Some
earlier applications to social, economical, and financial systems were reviewed
by Gorban, Smirnova, & Tyukina (2010).
The theoretical basis of these applications can be found in the quasistatic
theory of optimal resource allocation for different factors. It analyses the optimal
distribution of the total allocated AE between factors. In the previous sections
of our work we develop and analyse dynamical models of adaptation to one-
factor load. We have to go ahead and create the plausible dynamical model of
21

adaptation to multifactor load. It is very desirable to introduce as little new
and non-measurable details as possible.
Let us start from the models (6). First of all, we propose to use for the total
AE supply kr0(1 −W) instead of kr0(f−r). For one factor with the simplest
fitness function it is just redefinition of constant k←kψ0. Second, the AE
distribution should optimize Wand the simplest form of such an optimization is
the gradient descent. Immediately we get a simple system (perhaps the simplest
one) which is the direct generalization of (6) and follows the idea of distribution
of the resource between factors for fitness increase.
dri
dt=−kdr+kr0(1 −W)
∂W (ψ1,...,ψq)
∂ψi
Pi
∂W (ψ1,...,ψq)
∂ψi
;
dr0
dt=−kd0r0−kr0(1 −W) + krv Bo/c rrv (R0−r0) + kpr (R0−r0)W;
drrv
dt=−kd1rrv −krv Bo/c rrv (R0−r0) + kpr1(Rrv −rr v )W,
(16)
where ψi=fi−ri; changes of the Boolean variable Bo/c follow the rules formu-
lated above (see Fig. 6).
The fitness function should satisfy the following requirements: it is defined
in a vicinity of Rq
+, 0 ≤W≤1, W(0) = 1, ∂W/∂ψi≤0, gradW= 0 in Rq
+if
and only if W= 1, if ψi<0 then ∂W/∂ψi= 0.
The proposed model of the adaptation to the load of many factors needs
further analysis and applications. The well-studied quasistatic model appears
as a particular limiting case of (16) for slow degradation and fast resource re-
distribution.
The supply of AE to neutralization of each (ith) factor is in (16)
kr0(1 −W)
∂W (ψ1,...,ψq)
∂ψi
Pi
∂W (ψ1,...,ψq)
∂ψi
.
Here, the value of the factor at kr0is always between zero and one. In (1) and
(3 we used k0r0(f−r). This expression should be corrected by saturation at
large f−rbecause the rate of AE suply cannot be arbitrarily large: “there is
some upper limit to the amount of AE that an individual can use at any discrete
moment in time” (Selye’s Axiom 2). In (16) we get this saturation from scratch.
8. Conclusion and outlook
In this paper we aim to develop a formal interpretation of Selye–Goldstone
physiological theory of adaptation energy. This is an attempt at top-down
modelling following physiological ideas. These ideas were well-prepared by their
authors for formalization and were published in the form of ‘axioms’.
The hierarchy of two- and three- dimensional models with hysteresis is pro-
posed. Several effects of adaptation dynamics are observed as oscillations in
22

death or remission. These oscillations do not require any external reasons and
have intrinsic dynamic origin. Observation of such effect in the clinic was al-
ready reported for some diseases.
The dynamic theory of adaptation when the organism is subject to a load of
several factors needs further development. Goldstone (1952) formulated a series
of questions for the future dynamical theory of adaptation. More precisely,
there was one question and several apparently contradictory answers supported
by the practical observations:
“How will one stimulus affect an individual’s power to respond to a different
stimulus? There are several different and apparently contradictory answers; yet,
in different circumstances each of these answers is probably true:
1. If an individual is failing to adapt to a disease he may succeed in doing
so, if he is exposed to a totally different mild stimulus (such as slight fall
of oxygen pressure).
2. In the process of adapting to this new stimulus he may acquire the power
of reacting more intensely to all stimuli.
3. As a result of a severe stimulus an individual may not be able to adapt
successfully to a second severe stimulus (such as a disease). If he is already
adapting successfully to a disease this adaptation may fail when he is
exposed to a second severe stimulus.
4. In some diseases (those of adaptation) exposure to a fresh severe stimulus
may cure the disease. Exposure to an additional stressor will bring him
nearer to death but the risk may be justifiable if it is likely to re-mould
the adaptive mechanism to a normal form.
Future theoretic development should help to predict, which of these contra-
dictory answers will be true for a given patient. Currently we are still unable to
give such a prediction for individual patients but the quasistatic theory achieves
some success in predictions for groups and populations (Gorban, Manchuk, & Petushkova,
1987; Sedov et al., 1988; Karmanova, Razzhevaikin, & Shpitonkov, 1996; Pokidysheva, Belousova, & Smirnova,
1996; Svetlichnaia, Smirnova, & Pokidysheva, 1997; Vasi’ev et al, 2007; Razzhevaikin & Shpitonkov,
2008; Gorban, Smirnova, & Tyukina, 2010; Gorban et al., 2011; Censi et al., 2011;
Razzhevaikin & Shpitonkov, 2012; Bezuidenhout, van Antwerpen, & Berry, 2012;
Rybnikova & Rybnikov, 2012; Pokidysheva & Ignatova, 2013; Bernardini et al.,
2013). These authors proposed and tested a universal rule to investigate in
practice the amount of stress sensed by the system (and thus the danger of
catastrophic changes). The apparent universality of the top-down models of
adaptation could sometimes help in the solution of the important general prob-
lem of anticipation of critical transitions (Scheffer et al., 2012) and we should
also try to apply these models in general settings.
It is necessary to validate predictions of the models. Perhaps, some further
improvements are needed. For example, the classical description of the physio-
logical reaction to a noxious stimulus includes three phases (Selye, 1936): alarm–
resistance–exhaustion (the general adaptation syndrome, GAS). The alarm phase
could be described more precisely than it is done in the model (6) if we introduce
an activation threshold. One Selye’s axiom requires a threshold for activation of
23

the AE supply: “There is a threshold of AE activation that must be present to
potentiate an occupational response.” We introduced a threshold for the activa-
tion of reserve but did not use a threshold for the activation of the start of AE
supply (thus, in our models there are two levels of AE supply). Perhaps, such
a threshold of initial AE activation could help in the precise description of the
alarm phase. This threshold was even included by Chrousos & Gold (1992) in a
general definition of the stress system: “The stress system coordinates the gen-
eralized stress response, which takes place when a stressor of any kind exceeds a
threshold.” There is some empirical evidence of the existence of a hierarchy of
many activation thresholds (Garkavi et al., 1998). Construction of the models
with a hierarchy of thresholds does not meet any formal difficulty but increases
the number of unknown parameters.
Another improvement may be needed for the description of a dynamic re-
sponse of the instant fitness to changes of factors. In the proposed models,
the fitness reacts immediately. This seems to be an appropriate approximation
when the intensities of the factors change slowly but in a more general situation
we have to add a differential equation for the fitness dynamics.
There also remains a theoretical (or even mathematical) challenge: the sys-
tematic and exhaustive analysis of the phase portraits of the system (6) over
the full range of parameters.
Many data about physiological, biochemical, and psychological mechanisms
of adaptation and stress were collected during decades after Selye’s works (Chrousos & Gold,
1992; McEwen, 2007). The published schemes of the stress systems and regu-
lations include many dozens of elements. Mathematical models of important
parts of homeostasis have been created (Pattaranit & Van Den Berg, 2008). In
this situation, the simple models based on the AE production, distribution and
spending have to prove their usefulness.
The adaptation models introduced and analysed in this work exploit the
most common phenomenological properties of the adaptation process: home-
ostasis (adaptive regulation), price for adaptation (adaptation resource), and
the idea of optimization (for the multifactor systems). The developed models
do not depend on the particular details of the adaptation mechanisms.
These models, which are independent of many details, are very popular in
physics, chemistry, ecology and many other disciplines. They aim to capture
the main phenomena. In order to clarify the status of these models, we use the
classification of models elaborated by Peierls (1980). He introduced six main
types of models:
•Type 1: Hypothesis (‘Could be true’);
•Type 2: Phenomenological model (‘Behaves as if...’);
•Type 3: Approximation (‘Something is very small, or very large’);
•Type 4: Simplification (‘Omit some features for clarity’) ;
•Type 5: Instructive model (‘No quantitative justification, but gives in-
sight’);
24

•Type 6: Analogy (‘Only some features in common’);
At a first glance, we have to attribute our models to Type 4 or even to Type
5. Many famous models belong to these types: the Van der Waals model of
non-perfect gases, the Debye specific heat model (Type 4); the mean free path
model for transport in gases, the Hartree-Fock model for nucleus, and the Lotka-
Volterra model of predator-prey systems (Type 5).
Nevertheless, is seems to be possible to attribute the models of adaptation
elaborated in this framework of the top-down approach to the second or even
to the first type. Different biological systems that have evolved can have struc-
tures with analogous forms or functions but without close common ancestor
or with different intrinsic mechanisms. This is convergent evolution (McGhee,
2011). Some famous examples are: evolution of wings, eyes, and photosynthetic
pathways. The number of evolutionary pathways available to life may be quite
limited, and the functional response to the similar environmental challenges
may be similar without homology (no close common ancestor) and even with
different mechanisms.
Adaptation is a universal property of life and there are many mechanisms of
adaptation. Different detailed mechanisms may produce the same phenomeno-
logical answer at the top level because of convergent evolution. Let us call
this hypothesis the Principle of phenomenological convergence. The term ‘phe-
nomenological convergence’ was used in the analysis of synthetic biology by
Schmidt (2016) (phenomenological convergence of nature and technology).
The principle of phenomenological convergence results in the conclusion that
the general dynamic properties of adaptation may be much more universal than
the particular biochemical and physiological mechanisms of adaptation. This
manifested independence of the top phenomenological level from the bottom
level (detailed mechanisms) is the result of convergent evolutions. This allows
us to use AE models without solid knowledge of the intrinsic mechanism (behave
as if it is true, Type 2) or even to accept them as the truth (temporarily, of
course, Type 1).
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