3 Optimal Control, Variational Methods, and Multienzymatic Kinetics
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OC metabolic engineering to optimize metabolic regulation in vivo, from the perspective of minimizing metabolic expenses for an optimal substrate/product output
regulation, and thermodynamic optimization by processing at the minimum of thermodynamic potentials. In this sense, glycolysis is one of the basic metabolic pathways, and further studies are required to revise and extend the understanding of
optimal controllability, comparing results with alternative regulation approaches
where appropriate.
It is well known that enzymes are structures that affect the rate of chemical reactions without shifting the thermodynamic equilibrium [91]. The models in this
section employ the glycolysis model as one of the basic models (good working
examples) in metabolic network kinetics. The goal here is to illustrate that the spectrum of dynamical behavior after the introduction of optimal control into this kinetics does not change the topology of the main, metabolically sensible and steady/
equilibrium state taking place within the system without explicit control. Optimal
control interpretation of the extended model in terms of metabolic/catabolic costs/
losses is another area of focus in this section.
3.3.2
Optimal Control Introduction into the Bier-Teusink- KholodenkoWesterhoffÀVolkenstain Model of Glycolysis
The results in previous sections encourage one to apply the proposed approach to study
a more complicated system. It would be interesting to consider a well-investigated
pathway of biochemical reactions, and when the behavior in this pathway is imposing
the requirement of optimal control on the pathway regulation. A good example could
be the model of glycolysis, well investigated from many aspects including biochemical, evolutionary, and mathematical. It is also known that glycolysis contains more
than 20 intermediate stages [99,114]; some of them are shown in Figure 3.21.
The system of kinetics equations, first proposed by Higgins [115] and Sel’kov [116],
can be taken as a formal mathematical model of glycolysis. This model was modified
by Bier et al. [117,118] for the glycolytic dynamics in a yeast cell to the following
dynamic system. Different modification was also described by Volkenstain [119]:
x_ 5 Vin 2 k1 xy;
y_ 5 νk1 xy 2 kp
y
;
KM 1 y
ð3:94Þ
where x is the concentration of glucose (fructose-6-phosphate), y is the ATP concentration, Vin represents the constant inflow of glucose, k1 represents the phosphofructokinase activity, KM is the Michaelis constant for pyruvate kinase (PK), ν is
the stoichiometric ratio; ν 5 1 in the system described by Volkenstain [119] and
ν 5 2 for the model described by Bier et al. [117,118], the BTKW model. In
Figure 3.22, the numerical solutions are illustrated for the Volkenstain and the Bier
systems (with parameters described in Ref. [118]). One can see topological identities of these two models; therefore, we will designate as the Bier et al.À
Volkenstain (BTKW-V) model.
Variational and the Optimal Control Models in Biokinetics
Glc
ν1
FBP
ν2
NAD+
TP
ν3
NADH
BPG
NADH
2 ATP
2 ADP
157
2 ADP
ν7
NADH
ν4
2 ATP
NAD+
Pyr
ν5
NAD+
EtOH
ν6
Figure 3.21 Metabolic glycolytic pathway: Glc, glucose; TP, three phosphoglycerate; NAD+ ,
nicotinamide adenine dinucleotide; NADH, reduced form of NAD+; ADP, adenosine
diphosphate; ATP, adenosine triphosphate; BPG, biphosphoglycerate; FBP, fructose-1,6bisphosphoglycerate is inhibited by a reaction of phosphofructokinase; combined with
hexokinase, which needs ATP. This reaction is incorporated in v1. Pyr, pyruvate; EtOH, ethanol.
(A)
(B)
Figure 3.22 The numerical solutions of system (3.94) in double-logarithmic coordinates.
(A) The system described by Volkenstain [119] (stoichiometric ratio ν 5 1); (B) model
described by Bier and coauthors [117,118] (stoichiometric ratio ν 5 2). KM 5 2, k1 5 0.5,
kp 5 3.5, Vin 5 0.250. Curve “1”, x(t0) 5 9.0, y(t0) 5 0.5.
3.3.3
Direct Optimal Control Outline
To study the effect of the OC implementation with respect to the control of system
behavior, we used the Pontryagin maximum principle in the way it was used within
the MichaelisÀMenten system (Section 3.2). Let us formulate the problem of the
introduction of optimal control into the BTKW-V model (3.94) by k1 and kp. Then
the constraint system of equations will be
x_ 5 Vin 2 ðk1 2 uÞxy; xðt0 Þ 5 x0 ;
y
;
y_ 5 2ðk1 2 uÞxy 2 ðkp 2 vÞ
KM 1 y
yðt0 Þ 5 y0 :
ð3:95Þ
Let us consider the optimal control problem for this system, taking into account
the metabolic losses for control T(u, v) and the metabolic losses for not being in
thermodynamic steady/equilibrium state as G(x, y). The minimizing functional will
be similar to Eq. (3.84):
ðτ
ðGðx; yÞ 1 Tðu; vÞÞdt-min;
ð3:96Þ
t0
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The Common Extremalities in Biology and Physics
and the Hamiltonian function
H 5 2Gðx; yÞ 2 Tðu; vÞ 1 px ðVin 2 ðk1 2 uÞxyÞ 1 py 2ðk1 2 uÞxy 2 ðkp 2 vÞ
y
:
KM 1 y
ð3:97Þ
Then the canonical system will be
@H
5 Vin 2 ðk1 2 uÞxy; xðt0 Þ 5 x0 ;
@px
@H
y
; yðt0 Þ 5 y0 ;
5 2ðk1 2 uÞxy 2 ðkp 2 vÞ
y_ 5
@py
KM 1 y
@H
@G
5
1 px ðk1 2 uÞy 22py ðk1 2 uÞy;
p_x 5 2
@x
@x
@H
@G
KM
;
5
1 px ðk1 2 uÞx 22py ðk1 2 uÞx 1 py ðkp 2 vÞ
p_y 5 2
@y
@y
ðKM 1 yÞ2
x_ 5
ð3:98Þ
@H
@T
52
1 px xy 2 2py xy 5 0;
@u
@u
@H
@T
y
5 0:
52
1 py
@v
@v
ðKM 1 yÞ
By using the last two equations, it is possible to reduce the canonical system to
four equations—for two state and two co-state variables. For the square-law cost
for the deviations of the constants k1 and kp from optimal k1 and kp and square-law
form of G(x, y) and T(u, v), the following simplified expressions can be used:
Gðx; yÞ 5
kðx 2 xeq Þ2
kðy 2 yeq Þ2
1
;
2
2
Tðu; vÞ 5
u2
v2
1 :
2
2
ð3:99Þ
Finally we can obtain the system
x_ 5 Vin 1 ðpx 2 2py Þx2 y2 2 k1 xy; xðt0 Þ 5 x0 ;
kp y
py y2
; yðt0 Þ 5 y0 ;
2
y_ 5 22ðpx 2 2py Þx2 y2 1 2k1 xy 1
ðKM 1 yÞ2 KM 1 y
@G
2 xy2 ðpx 2 2py Þ2 1 k1 yðpx 2 2py Þ;
p_x 5
@x
p_y 5
ð3:100Þ
p2y KM y
K M k p py
@G
1 x2 yðpx 2 2py Þ2 2
1 k1 ðpx 2 2py Þx 1
:
@y
ðKM 1 yÞ3
ðKM 1 yÞ2
Numerical solutions of the system show the existence of the torus-like steady
near former two-dimensional limit cycle. Figure 3.23 shows the trajectories for
state variables x, y; momenta px, py (co-state variables); and control u, v for system
Variational and the Optimal Control Models in Biokinetics
159
Figure 3.23 Trajectories for
state variables x, y; momenta
(co-state variables) px, py; and
control u, v for system (3.100)
using (3.99). x(t0) 5 1.5, y
(t0) 5 0.5, px(t0) 5 py(t0) 5 2
0.01, Vin 5 0.250, KM 5 2,
k1 5 0.5, kp 5 3.5, Vin 5 0.250,
potential (k 5 0.01).
(3.100) using the potentials from (3.99). Figure 3.24 shows the numerical solutions
in this case at the values of constants, specified directly in the figures.
The limit cycle phase plots, which characterize the six-dimensional phase space
of two state variables, two co-state variables, and two control variables, altogether
six variables, are illustrated in Figure 3.24 for the system (3.100) at square-law
potential. The plots are shown as a graphical matrix, where just half of it, as it is
shown in the figure, can fully characterize the six-dimensional limit cycle. The first
row contains two-dimensional graphs for the vertical x coordinate versus all other
coordinates (e.g. y, px, py, u, v) spanned horizontally. The second row contains vertical (y) coordinate against px, py, u, and v; y against x already is plotted in the first
row. Effectively, the combination of any pair of coordinates can be found using the
designations for the correspondent row (at first left plot in the row) and for the correspondent column (at the bottom at any column). For example, the top row and
right column illustrate the phase plot in the xÃ v diagram.
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The Common Extremalities in Biology and Physics
Figure 3.24 Graphical matrix characterizing the behavior of the limit cycle (3.100) in the
six-dimensional phase space by projection of it into all combinations of two-dimensional
planes. The red trajectories are for starting values x(t0) 5 1.5, y(t0) 5 0.5. Blue curve, for
x(t0) 5 4.0, y(t0) 5 0.1. Constants were chosen following BTKW-V model [117,118];
Vin 5 0.250, k1 5 0.5, kp 5 3.5; KM 5 2.0.
The effect of a number of constants is illustrated in Figure 3.25 for system
(3.100); the phase plot for control variables is not included. The plots are shown as
matrices, similar to Figure 3.24.
As one can see from Figure 3.25A, an increase in Vin in the range shown in
figure reduces the limit cycle. A span in the value of xeq through the equilibrium
point of the system (3.100) leads to the collapse of the limit cycle (Figure 3.25B).
Similarly, the span in yeq through the equilibrium point of the system also leads to
the collapse of the limit cycle (Figure 3.25C). Figure 3.25D indicates that the
increase in KM (inhibition) leads to a reduction in the size of the limit cycle.
However, a simultaneous change of the parameters of the system (3.94/3.100) in
Variational and the Optimal Control Models in Biokinetics
161
the effect of the robustness (stability of the equilibrium points of the system) will
be illustrated further in this section.
It can be useful to illustrate the comparison of the effects on the limit cycle
in normal and logarithmic coordinates (Figure 3.26). Although the normal coordinates clearly show what is happening at large values of the state variables
(Figure 3.26A), the logarithmic coordinates (Figure 3.26B) efficiently illustrate
what is happening at the very low values of the state variables. Particularly, for the
systems (3.94) and (3.100), one can see that implementation of the optimal control
and demonstrating that a quadratic form of the penalties does not significantly
change the character and robustness of the limit cycle. However, by using the
standard method, we are more limited in illustrating the effects of the parameters on
the character of stability in a wide range of the variables’ and parameters’ values.
3.3.4
Variational Formulation
We can apply the method developed in Refs. [67,68] to formulate a pure variational
approach. From Eq. (3.95), we can find controls u and v:
u 5 uo 1
x_ 2 Vin
;
xy
ð3:101aÞ
v 5 vo 1
ðy_ 1 2x_ 2 2Vin ÞðKM 1 yÞ
y
ð3:101bÞ
and substitute them into the Lagrangian from Eq. (3.96), taking the penalty for not
being in a steady/equilibrium state G(x, y) and the cost of control T(u, v), which
can be associated with metabolic and energetic losses, in quadratic form (3.99).
Then we obtain the variational Lagrangian
1
ðy_ 1 2x_ 22Vin ÞðKM 1 yÞ 2
x_ 2 Vin 2 1
_ yÞ
_ 5 Gðx; yÞ 1
1
:
k1 1
kp 1
Lðx; y; x;
xy
2
2
y
ð3:102Þ
From the Lagrangian, one can find the EulerÀLagrange equations.
It is also possible by applying the Legendre transform:
@L
1
2ðKM 1 yÞ
ðy_ 1 2x_ 22Vin ÞðKM 1 yÞ
x_ 2 Vin
5
k1 1
kp 1
;
1
@x_
xy
y
y
xy
ð3:103aÞ
@L ðKM 1 yÞ
ðy_ 1 2x_ 22Vin ÞðKM 1 yÞ
py 5
5
vp 1
ð3:103bÞ
@y_
y
y
px 5
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The Common Extremalities in Biology and Physics
(A)
(B)
Figure 3.25 The phase plot of the limit cycle illustrating the effect of different constants in
the system (3.100) on the limit cycle. (A) The effect of Vin (green color, Vin 5 0.1; blue,
Vin 5 0.2; red, Vin 5 0.3); other constants (k 5 0.01, xeq 5 1.3, yeq 5 0.2, k1 5 0.5, veq 5 3,5,
(C)
(D)
Figure 3.25 (Continued)
KM 5 2.0). (B) The effect of xeq (green color, xeq 5 1.2; blue, xeq 5 1.5; red, xeq 5 2.0); other
constants (Vin 5 0.1, k 5 0.01, yeq 5 0.2, k1 5 0.5, veq 5 3,5, KM 5 2.5). (C) The effect of yeq
(green color, yeq 5 0.01; blue, yeq 5 0.1; red, yeq 5 1.0); other constants (Vin 5 0.1, k 5 0.1,
xeq 5 1.5, k1 5 0.5, veq 5 3.5, KM 5 2.5). (D) The effect of KM (green color, KM 5 1.8; blue,
KM 5 2.0; red, KM 5 2.5); other constants (Vin 5 0.1, k 5 0.01, xeq 5 1.5, yeq 5 0.2, k1 5 0.5,
veq 5 3.5). (For interpretation of the references to color in this figure legend, the reader is
referred to the web version of this book.)
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The Common Extremalities in Biology and Physics
(A)
(B)
7
10
3
1
–1
–0.1
0.6
1.3
2
0.1
1·10–5 1·10–4 1·10–3
0.01
0.1
1
10
Figure 3.26 The comparison of the limit cycle in the phase plane of state variables for
different systems. (A) Comparison in normal coordinates; (B) the limit cycles in the
logarithmic coordinates. Cyan curve—original system (3.94) without OC; magnolia curve—
system (3.100) with the OC introduction, no potential term (k 5 0), initial momenta
(px 5 py 5 0.0); green curve—system (3.100) in the presence of OC, no potential term
(k 5 0), starting momenta px 5 py 5 20.5; blue curve—OC, potential (k 5 0.01), initial
momenta (px 5 py 5 0.0); red dashed curve—OC, potential term (k 5 0.01), initial momenta
(px 5 py 5 20.5). Constants were chosen following Bier et al. [117,118]; Vin 5 0.10,
k1 5 0.5, kp 5 3.5; KM 5 2.0. (For interpretation of the references to color in this
figure legend, the reader is referred to the web version of this book.)
to find variational Hamiltonian
p2y y2
1
ðpx 22py Þ2 x2 y2 1
2
ðKM 1 yÞ2
0
1
k
y
p
A:
1 px ðVin 2 k1 xyÞ 1 py @2k1 xy 2
ðKM 1 yÞ
Hðx; y; px ; py Þ 5 2Gðx; yÞ 1
ð3:104Þ
This Hamiltonian gives the same system of canonical equations as Eq. (3.100).
Therefore, the EulerÀLagrange equations and the canonical equations can be written for the Bier model.
3.3.5
Statistical Method to Study the Robustness
To study the character of the equilibrium points in a wider range of parameters,
one can span these parameters in a range, keeping all others at certain values,
as illustrated in Figure 3.27. One can also see the existence of a limit cycle (designated as “o”) in a wide range of parameters. However, spanning of a single parameter does not show the complete picture of the parameter’s role in the equilibrium.
Variational and the Optimal Control Models in Biokinetics
(B)
102
k1
(A)
165
100
Vin
101
10–2
100
100
100
Y
10
0
–3
10
10
–2
Y
10–1
X
100
10–2
(C)
X
(D)
102
K
kp
101
101
100
100
10–1
–1
10–2
10
100
Y
10
10–2
0
X
102
101
102
0
Y 10
100
X
10–1 10–2
Figure 3.27 The illustration of parameter’s spanning in model (3.94) at Vin 5 0.25, k1 5 0.5,
kp 5 1.0, KM 5 1. (A) scanning Vin; (B) scanning k1; (C) scanning kp; (D) scanning KM.
On the other hand, the stability of the system can be studied in the following
way: One can generate a set of random combinations of all the parameters (Vin, k1,
kp, and KM) employing the MonteÀCarlo method [120] for statistical investigation
of the equilibrium points [121] of this system (3.94), with the purpose of comparing
to the system extended by applying the optimal control. The MonteÀCarlo simulation results are shown in Figure 3.28 with a range of parameters (Vin, k1, kp, and
KM) that are much more informative. From Figure 3.28, one can see main characteristic states (the designations are shown in Figure 3.28F). One can also clearly see
borders between the main areas, which indicate the transitional surfaces between
areas of different types of equilibrium. Results obtained by this method can be considered to be in good agreement with analytical results from Bier et al. [117,118].
It is well known that different equilibrium scenarios follow from the spectrum of
eigenvalues of the Jacobean matrix at an equilibrium state (see Figure 3.28F for
a four-dimensional system). The transitions between qualitatively different states
(bifurcations) are also of immense interest because they indicate qualitative changes
in the system dynamics, suggesting that a closer look of the robustness and equilibrium of the system is needed. It is evident that the scatterplots, shown in Figure 3.29,
qualitatively reproduce the plots for the reduced Bier model (Figure 3.28); however,
the difference is in the character of stability. This is not obvious because the model is
significantly changed, as can be seen from Eqs. (3.94) and (3.100). One can see that
with different fixed levels of potential, Figure 3.29 (B, k 5 0; C, k 5 0.1; and D,
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The Common Extremalities in Biology and Physics
k 5 1.0), the three-dimensional scatterplot of the dataset becomes less transparent
(Figure 3.29BÀD), in the sense of the different character of equilibrium areas.
Figure 3.30A shows that when the MonteÀCarlo method is applied to randomly generate the values of the variable k (potential weight) and KM, the scatterplot of the
dataset is not transparent. However, by applying the canonical correlation analysis
(CCA) [122] method to the dataset, it is possible to distinctly observe the regions of
stability in the plane of the first two canonical variables (Figure 3.30B).
In some sense, conventional CCA generalizes the principal component analysis
(PCA). A good review on nonlinear PCA and its further development can be found
in Refs. [95,123À129].
In Table 3.1, the results of applying of the MonteÀCarlo method that we used
to study the character of stability of the equilibrium points (x, y) for system (3.94)
show the impact of the system parameters on stability. About 10,000 combinations
of parameters in the range (see Figure 3.29) were generated. The first three eigenvalues obtained were 0.8345, 0.0766, and 0.003 with F-statistic values 669, 121,
and 11, respectively, and the probability levels less than 0.0001 each. As it is
shown in Table 3.1, the raw canonical coefficients for the first canonical variable,
Can1, indicate that the classes differ most widely on the linear combination of the
centered variables 1.202Ã Vin 1 0.101Ã k1 2 0.26Ã kp 1 0.38Ã KM. Therefore, the results
indicate the crucial role of Vin on the character of stability.
Results in Figure 3.29 relating to the existence of different areas of stability
qualitatively agree with the numerical results for system (3.94), which is without
the OC (Figure 3.28). The cooperative form indicates the goal of the optimal adaptive regulation that in the metabolic network could strengthen the rigidity of the
regulation around the macroscopically important state.
To study the stability character of the points of equilibrium (x, y, px, py) of
the system (3.100), when the control variable u and v are eliminated, the
MonteÀCarlo method was also employed. Five thousand random combinations
of parameters were generated. The first four eigenvalues were 0.86, 0.11, 0.04, and
0.005, with the Fisher statistics F-values 242, 67, 35, and 8.9, with the corresponding probabilities less than 0.0001 each. As is evident in Table 3.2 for the raw
canonical coefficients, the first canonical variable, Can1, shows that the linear
combination of the centered variable Can1 5 2.234Ã Vin 2 0.055Ã xeq 2 0.005Ã yeq 1
0.019Ã k1 2 0.308Ã kp 1 0.290Ã KM 1 0.554Ã k separates the areas with different characters of stability most effectively.
3.3.5.1 Optimal Control by KM in the BTKW Model of Glycolysis
The optimal control implementation into the Bier et al. model [117,118] of glycolysis by KM can be done in a similar way, by replacing KM with control u in
Eq. (3.94) and applying functional in the form
ðτ
ðGðx; yÞ 1 TðuÞÞdt-min
t0
ð3:105Þ