4 Optimal Control in Hierarchical Biological Systems: Organism and Metabolic Hierarchy
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The Common Extremalities in Biology and Physics
ordered systems of these molecular and chemical reactions, located on and separated by membranes with a total surface area of thousands of square meters.
Organisms are, however, chemical machines, which have acquired, during their
evolution, a huge number of degrees of freedom with complexity throughout the
entire hierarchy. Consequently, in such systems, neither the reactions nor their products are strictly determined and rigidly interrelated; their interinfluence is mediated and not so rigidly determined as in vitro, in a probe or in a chemical reactor;
this suggests a nonrigid organization. Such nonrigid relationships among parameters may often grant higher adaptive properties to a whole system and could
play a major role in competition, enabling the system to adapt to conditions in a
quickly and persistently varying environment. The biotic cell is a good example of
such a nonrigid system, which has existed for more than 3 billion years; perhaps
none of the rigid systems have existed on Earth for such a long time.
A characteristic property of the nonrigid systems is that under continuously varied external conditions, the very important parameters (macroparameters) of the
system vary only slightly, although the system as a whole can deviate substantially
from the initial starting state on another microscopic level. The system can remain
in a physiologically acceptable, macroscopically optimal state (it is, therefore, the
macroscopic conditions that determine the competitiveness in the most explicit
way), although the concentrations of some metabolites can vary substantially. So,
one can say that there are some parameters that are maintained by a biotic system
in a target-oriented way, and it is this maintenance that constitutes the adaptive
strategy. Such parameters are effectively new degrees of freedom that cannot be
reduced to a set of concentrations themselves. Examples are such physiological
parameters as temperature, blood pressure, oxygen concentration, and so on.
Practically, as we have stressed in previous sections, in the majority of cases the
difference between physiological, metabolically normal, steady states in the biological system and the states that are far from being optimal can be characterized by a
vector. This vector will contain some measurable values as its components—those
that are characteristic of the system from the perspective of experimentation. The
problem of evaluating this metabolic difference is sometimes complicated, and
intensive use is made of data processing techniques to help. Such mathematical
tools and procedures are provided, for example, by multidimensional statistical
analysis; these procedures, however, are not always as simple as Student’s t-test or
even the Bonferroni corrections for multiple comparisons [135].
In its macroscopic versus microscopic presentations, the hierarchy of values
describing biological objects seems to be more sophisticated than in physics. We
should particularly emphasize the adaptive characteristic of the variability of
biological parameters. The measured intensity of a biological parameter (macroscopic or microscopic) depends on the state (e.g., activity levels and level of
intoxication) of this biological object. As a consequence of a biosystem’s ability
to exist in different states, there can be a certain variability in its macroscopic
(physiological in the case of an organism) characteristics, determined by the state
of the environment, and microscopic parameters, relating to the internal regulatory state of the cells and tissues. We can, therefore, see that the majority of
Variational and the Optimal Control Models in Biokinetics
175
biological processes manifesting satisfactory adaptation occur at a macroscopic
level. Therefore, in measuring metabolic levels within a biological system, we
need to recognize that all macroscopically and microscopically measured parameters have three distinct terms: average level, variability related to functional
range, and errors in experimental measures. On this basis we find that biological
parameters have a significant range of variability and the most variable part can
be determined within the framework of one species by looking at one individual.
By studying this variability within a sample (general population), we can characterize the regulatory deviation of the system from the normal or optima state and
elucidate the mechanism of control of the system at a specific moment in the
adaptive process.
At the same time, the number of biological parameters, which can be measured
in the body or even in a more simple metabolic system, a cell, is enormous. Such a
set of parameters used to characterize the state of a biological object is a vector,
the dimension of which coincides with the amount of possible parameters measured. The sheer scale of possible parameters involved in a biological system
means that it is necessary to reduce the number of dimensions in order to make any
model of the system more transparent. Such a reduction must, however, minimize
the loss of information about the system, while maintaining the ability to interpret
adequately any changes in the system. Such methods are often referred to as information reduction.
Such methods could potentially be used to find the “key control points” in metabolic networks of substances and enzyme concentrations within a cell or organism.
In other words, this method could allow us to solve the problem that previous deterministic schemes of study of the metabolic control of every substrate were unable to.
One of the most well-known statistical methods of study of such multidimensional states is the method of principal components. It should be noted that this
method allows not only statistical interpretation, but borders upon statistical and
deterministic methods.
The characteristic feature of this method is finding the linear combinations or
the principal components that are, in some sense, the most informative combinations. Establishing the topography of physiological and biochemical states in the
plane of the principal components, as well as finding the trajectories of these states
in the plane, provides important information in determining the measure of normalization of the state and the degree of dynamic equilibrium attained in the targetoriented correction of toxic states. In this situation, the regulation could be not
simply reduced to control by means of individual components of the measured vector, but by an additive set, which can be related to the first or the second principal
component. Therefore, we can recognize that this method is a potentially useful
way of revising the problem of the introduction of optimum control on the basis of
the formulation of the cost to the organism of metabolic control and the cost of
deviation from the equilibrium in terms of the principal component.
Here, we offer a short outline of the principal components method. Let xj be the
components of the vector, which characterize the state of a system, but the dimension
of this vector is redundant. Therefore, we will search for such linear combinations Yi
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The Common Extremalities in Biology and Physics
of the initial component xj for which would have smaller dimension, but characterized
the system well, i.e., the transition from xj to new Yi (see, for example, Pearson [130]):
Yi 5
p
X
αij xj :
ð3:108Þ
j51
Thus, we should say about the existence of sufficiently strong restrictions for the
covariance matrix and variance, requiring noncorrelation of the new variables Yj:
CovðYi ; Yj Þ 5 0; i; j 5 1; 2; . . . ; p; i 6ẳ j;
3:109ị
and ordering of ls component on increasing the variance:
VðY1 Þ $ VðY2 Þ $ ? $ VðYp Þ:
ð3:110Þ
Moreover, the total variance after transformations should remain without changes:
p
X
i51
VðYi Þ 5
n
X
σj :
ð3:111Þ
j51
Sometimes the subset of the few (2À4) first variables Yj could explain the large
part of total variance and, therefore, a satisfactory description of the structure of the
dependence of the initial variables will be obtained. A good further generalization
of this method, known as invariant/principal manifolds, one can find in works of
Gorban and co-authors [95,123À129].
Two interesting examples of vectors that can be used to describe the state of a
system like a human organism are the free amino acid pool in the blood or a tissue
and the level of steroid hormones. In the first case this vector has a dimension of
20À30, in the second can be about 10À30.
It is possible to consider as an example the relaxation kinetics of the amino acid
pool to the normal physiological state in a plane of the first two principal components during an administration of coenzyme A [131]. A relaxation trend to the
control group (a group without any administration) state is seen clearly. If we represent this relaxation by the dependence of the first principal components on time,
we can see the obvious tendency to some norm. By performing exponential regression on the experimental points, one can see that the statistical significance of the
two-exponential regression for the first principal components is higher than for the
most informative, most variable amino acid—alanine. This could be explained by
complex reorganizations inside the amino acid pool and their metabolites during
the relaxation process. Comparison to other amino acid relaxation curves shows
that only the first principal component reflects the steady tendency to relaxation,
Variational and the Optimal Control Models in Biokinetics
177
whereas the other metabolites reflect the complexity of the accompanying metabolic perturbations.
The minimization trend of the deviation from the stationary state is more strictly
pronounced for the first principal component. This emphasizes that the organism is
a rigid system not for individual microparameters, but rather for some generalized
parameters. The principal components thus make the multidimensional space of
concentrations more transparent in the sense of the strategy of regulation and optimal control.
Furthermore, it is interesting to formulate the problem of optimal control
through the regulation of the relaxation kinetics of the system by the principal
components. We can try to deliver this problem formally—to formulate the metabolic penalty during the deviation of the organism from an optimal state through
the principal components Yi or through p, the first principal component. Let the
penalty for staying in the state, different from the optimal one, be in a square-law
form
p
X
βi
i51
ðYi 2 Yi0 Þ2
:
2
ð3:112Þ
If u is the vector of control, the cost of control for the organism can also be
chosen in square-law form
K u2
X
j
j51
2
:
ð3:113Þ
Then the minimized functional will be
J5
!
P
X
ðYi 2 Yi0 Þ2
uj 2
1
βi
dt-extr:
2
2
i51
j51
ðT X
P
0
ð3:114Þ
If the control u is (defined by some dynamical model):
dYi
5 fi ðY1 ; Y2 ; . . . ; YP ; u1 ; u2 ; . . . ; uK Þ;
dt
ð3:115Þ
then the corresponding Lagrange function will be
L5
P
X
i51
βi
P u2
P
X
X
ðYi 2 Yieq Þ2
j
1
1
λi ðY_ i 2 fi ðY1 ;Y2 . . . YP ;u1 ;u2 . . . uK ÞÞ:
2
2
j51
i51
ð3:116Þ
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The Common Extremalities in Biology and Physics
It is also possible to write the EulerÀLagrange equations:
P
X
@L
@fi
0
5 Yi 2 Yieq 2
λi
5 λi ;
@Yi
@Yi
i51
P
X
@L
@fi
5 ui 2
λi
5 0;
@ui
@ui
i51
@L
5 Y_ i 2 fi ðY1 ; Y2 ; . . . ; Yp ; u1 ; u2 ; . . . ; uK Þ:
@λi
ð3:117Þ
The given problem in a general case is rather faceless and can be interesting
only in some special cases.
In some sense, the principal components can play the role of parameters of the
order, similar to those in the models of Haken [132]. From the parameters of order,
another dynamic system could be designed, describing a new hierarchical level
[133,134], and newer hierarchies can be created through bottlenecks in the previous
level of regulation.
Therefore, a nonrigid, physiological level of regulation allows a conceptual and
technical formulation in terms of optimal control. This suggests that biosystems
functioning on different levels are involved in activity on different levels, and an
organism is a sublevel of the species form of biosystems presentation. One could
formulate a question: how does a similar optimal control formulation look in the
case of postorganismic, i.e., biocenotic and social systems, for which the kinetics is
manifested as the evolution?
We would like to emphasize one principal strategy in the regulation of biosystems—
nonrigid relationships exist between microscopic, molecular components in a biotic
system. In a metabolic system, therefore, it is impossible, and not necessary, to rigidly
control the concentration of all metabolites and activities. For faster and optimal control, it is necessary to operate only on “the key control points” instead of operating on
the whole number of parameters in a biosystem. It is enough to administer only important sites of metabolic pathways. The inherent multidimensionality incorporated in the
biosystems acquires even greater importance due to the materialization of the extreme
free energy dissipation.
In fact the increase of nonrigidity is an increase of the dynamic range of adaptability of a biotic system due to a decrease of the metabolic losses of regulation, as
the adaptation and its consequence—survival of a species is a form of strategy of
the species as an accelerating way of free energy dissipation.
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