Integrative Cancer Science and Therapeutics
Research Article
ISSN: 2056-4546
Phase transitions in tumor growth VII: The effect of
periodic glucose pulses and privations in a cancer model
Pomuceno-Orduñez JP1, Silva E1, Martin RR2, Durán I1, Bizzarri M3, Cocho G4, Mansilla R5 and Nieto-Villar JM1*
1
Department of Chemical-Physics, A. Alzola Group of Thermodynamics of Complex Systems of M.V. Lomonosov Chair, Faculty of Chemistry, University of
Havana, Cuba
2
Instituto de Investigaciones en Tecnología Química (INTEQUI), UNSL - CONICET, Argentina
3
Department of Experimental Medicine, Systems Biology Group, University La Sapienza, Italy
4
Departamento de Sistemas Complejos del Instituto de Física de la UNAM, Mexico
5
Centro de Investigaciones Interdisciplinarias en Ciencias y Humanidades, UNAM, Mexico
Abstract
Tumor glycolysis is a self-organized nonlinear mechanism that can exhibit oscillations under certain conditions. Experimental evidence suggests that glycolytic
oscillations may give certain advantages to cancer, including resistance to therapies. A previously developed oscillating glycolysis model was modified with the
objective of theoretically evaluating the effect of pulses and periodic glucose deprivations in a frequency range that included values close to the autonomous frequency.
The dynamics of the model before the perturbation were characterized. Then the model was perturbed with periodic pulses and deprivations of glucose. The dynamics
obtained were characterized and compared with the model before the perturbation. The methods for the characterization of the dynamics were: stability analysis of the
steady state, stroboscopic analysis and Lempel-Ziv index; while the cellular energy charge of the simulations was evaluated by the normalized ATP/ADP ratio. The
results obtained indicate that periodic glucose pulses can lead to an increase in the energy charge. Surprisingly, sustained increases in glucose influx causes a decrease
in the complexity of glycolytic oscillations, but cause increase in the cellular energy charge. On the other hand, periodically depriving the tumor microenvironment
of glucose at high perturbation frequencies, regardless of the amplitude of perturbation, prevent the increase in the complexity of glycolytic oscillations and cause a
decrease in the cellular energy charge of tumor cells. This strategy can increase the efficacy of antitumor therapies.
Introduction
The most frequent metabolic alteration in most tumors is the high
consumption of glucose and the excretion of large amounts of lactate
even under aerobic conditions, a phenomenon known as the Warburg
effect [1]. The high glycolytic rate provides metabolic energy and the
biosynthesis precursors required for cell proliferation [2].
The increased glycolytic flux in tumor cells is due to the overexpression of glycolytic enzymes and transporters [3] induced by the
activation of various oncogenes and the Hypoxia Induced Factor (HIF1α) [4]. This regulation of metabolism has been related to an increase in
tumor aggressiveness [5]. Thus, the glycolytic phenotype plays a role in
the progression of neoplastic cells [6].
Tumor glycolysis can oscillate under certain conditions, which has
been demonstrated in cells that develop a hypoglycemic phenotype
[7-9]. This phenomenon is verified through temporary oscillations in
the concentration of glycolytic intermediates. Experimental evidence
suggests that these oscillations may give certain advantages to cancer,
including resistance to therapies [8]. In fact, it is known that periodic
behavior gives biological systems certain properties such as greater
adaptability, synchronization possibilities and greater resistance to
fluctuations and environmental perturbations [10].
It has been shown that the complexity of tumor glycolysis in
the hypoglycemic state is superior to a hyperglycemic state [11].
Experimentally, it has been proved that in hypoglycemic states the
tumor glycolysis exhibits an increase in the levels of glucose transporters
Integr Cancer Sci Therap, 2019
doi: 10.15761/ICST.1000301
1 and 3 (3,4 and 2,1-fold, respectively) and hexokinase I (2,3-fold)
compared to the hyperglycemic standard cell culture condition [3].
Furthermore, a correlation between increased glycolysis and tumor
resistance to chemo- and radiotherapy has been found [3,12]. These
studies suggest that there is a positive correlation between complexity
and resistance. In fact, resistance to external perturbation may increase
with increasing complexity in oscillations [13]. Therefore, more
complex tumor glycolytic oscillations can be associated with greater
resistance to external perturbations aimed at affecting this metabolic
pathway. This increases the robustness of the system. According to
Kitano, robustness is a property that allows a system to maintain its
function against internal and external perturbation [14].
Cancer glycolysis is a self-organized process far from thermodynamic
equilibrium [15]. This metabolic network is characterized by a
synchronization of individual reactions under a nonlinear dynamic
[16] that exhibits feedback loops through enzymatic regulation [17].
Under these conditions the system can exhibit oscillations [13].
These premises led us to consider the following hypothesis: the degree
of self-organization achieved by tumor glycolysis can be affected
*Correspondence to: JM Nieto-Villa, Department of Chemical-Physics, A.
Alzola Group of Thermodynamics of Complex Systems of M.V. Lomonosov
Chair, Faculty of Chemistry, University of Havana, Havana, Cuba, E-mail:
nieto@fq.uh.cu
Received: February 10 2018; Accepted: February 21, 2019; Published: February
23, 2019
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Pomuceno-Orduñez JP (2019) Phase transitions in tumor growth VII: The effect of periodic glucose pulses and privations in a cancer model
by the application of periodic glucose perturbations simulating a
chronotherapeutic regimen. This kind of perturbation could influence
the complexity of glycolytic oscillations and energy metabolism of the
tumor cell, which would affect its resistance to therapies.
Chronotherapy is defined as a temporarily adequate medical
treatment that takes into account the biological rhythm of the organism
and guarantees greater efficacy in its application [18,19]. The effect of
periodic pulses of glucose has been experimentally evaluated in yeast
[20,21], however to our best knowledge there are no such experiences
in oscillating tumor glycolysis. On the other hand, the effect of periodic
nutrient deprivation in cancer has been studied and encouraging
results have been obtained [22]. That study, being experimental, only
represents perturbation periods much longer than the autonomous
period of oscillating tumor glycolysis, which is of the order of tens of
seconds [9]. Mathematical modelling allows us to explore what could
happen if the perturbation and the tumor oscillation were of the same
order.
Recently, a mathematical model was proposed to describe
qualitatively the autonomous glycolytic oscillations found
experimentally in tumor cells [15]. The goal of this work is to evaluate
the effect of periodic pulses and deprivations of glucose at frequencies
near the autonomous frequency of oscillating tumor glycolysis.
The manuscript is organized as follows: In Section 2, we propose
modifications to the previous glycolytic model [15] and a mathematical
function for describing the perturbation. Section 3 is devoted to the
analysis of the autonomous system and to the characterization of the
perturbed model with periodic pulses and deprivations of glucose.
Section 4 comprehends the discussion of the results. Finally, some
concluding remarks are presented in Section 5.
Methods
A biochemical network model of glycolytic oscillations
The kinetic model proposed by Martin et al. [15] adapted to
hypoglycemic HeLa cells was modified. However, the basic mechanism
of the glycolytic pathway was maintained as illustrated in Figure 1.
It was obtained from the adjustment of the model proposed by [15].
Dashed lines indicate activation of enzymes by metabolites. Reaction
1: GLUT + HK + HPI; reaction 2: PFK-1; reaction 3: non-glycolytic
consumption of F1,6BP; reaction 4: ALDO + TPI + GAPDH + PGK
+ PGM + ENO; reaction 5: PYK, reaction 6: MCP (mitochondrial
consumption of pyruvate); reaction 7: LDH; reaction 8: ATPase.
Reaction 1:
The HK reaction was replaced by another one that includes in a
single step the effect of the glucose transporters expressed in HeLa
and the reactions catalyzed by HK and HPI. The proposed rate law is
described as:
V1 = C - k1[F6P]
where C represents a constant flux and can be interpreted as the transport
rate of glucose from the extracellular medium to the intracellular one;
while ki is a rate constant.
Reaction 2 (PFK-1):
F1,6BP was included as a product that activates the enzyme PYK.
The basic structure of the equation used by Martin et al. [15] which
considers the activation by product (ADP) of PFK-1 was maintained.
Unlike the previous model [15], it was considered that the rate law
depends on the F6P substrate. This is possible using the equation
proposed by Goldbeter and Dupont [23].
[ F 6 P ] F 6 P n −1 ADP n
1 +
1 +
Ks
Ka
Ks
V2 = Vm
n
n
L + 1 + [ F 6 P ] 1 + ADP
Ks
K a
Vm is the maximum rate of the catalyzed reaction, n denotes the number
of subunits of the enzyme, Ks is the dissociation constant of the substrate
to the relaxed state of the enzyme, Kα is an activation constant and L is
the allosteric constant [24].
Reaction 3
A reaction representing the consumption of F1,6BP was added
through a non-glycolytic mechanism, which agree with previous works
[25]. This was done in order to avoid a sustained increase of F1,6BP.
V3=k3[F1,6BP]
Reaction 4:
The GAPDH reaction was replaced by another one that includes
in a single step the effect of the reactions catalyzed by ALDO, TPI,
GAPDH, PGK, PGM and ENO. The flux for this step was modified
to a mass action law. To achieve the characteristic carbon balance
of glycolysis, it was taken into account that for each mole of F1,6BP
consumed, two moles of PEP are formed. To simplify the equation, the
presence of inorganic phosphate and the NAD+/NADH system was not
considered, as had been done in previous works [26].
V4=k4[F1,6BP]([ADP])2
Reaction 5 (PYK):
The rate law of the reaction catalyzed by PYK was replaced by the
one proposed by Dynnik and Sel’kov [25] which considers the activation
of this enzyme by F1,6BP. This guarantees the coupling between PYK
and PFK-1. The rate law is described as:
V5=k5[PEP][ADP](z+([F1,6BP]/Ka)γ)
where γ is the order of activation of the enzyme by F1,6BP; z << 1 is
a dimensionless parameter that determines the relative activity of the
enzyme to [F1,6BP] = 0 [25].
Pyr consumption reactions
Figure 1. Representation of the kinetic model of glycolysis, adapted to HeLa tumor cells
Integr Cancer Sci Therap, 2019
doi: 10.15761/ICST.1000301
MCP and LDH represent the consumption of Pyr in the
mitochondria and the cytosol respectively. The first is represented by a
flux constant and the second by a mass action law [26].
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Pomuceno-Orduñez JP (2019) Phase transitions in tumor growth VII: The effect of periodic glucose pulses and privations in a cancer model
These values were obtained experimentally from HeLa cells adapted to
hypoglycemic conditions [3].
V6=C1
V7=k7[Pyr]
where C1 represents a constant flux.
Reaction 8 (ATPase)
The kinetics of the reaction catalyzed by ATPase defined as a process
of cellular consumption of ATP was taken as a law of mass action, which
coincides with Termonia and Ross [26] and Marín et al. [3,27].
V8=k8[ATP]
The consumption and production of ATP was adjusted in reactions
1, 2 (PFK-1), 4 and 5 (PYK), to yield a gain of two moles for each
inverted mole, as it occurs in the glycolytic pathway. This constitutes an
autocatalytic element that can lead to oscillations [28]. Table 1 shows
the values of the parameters of each rate law used.
proposed value; (b) value of L corresponds to a region of instability
of the steady state in the curve: nH (Hill coefficient) vs L, when n=4 (c)
values selected so that the LDH flux >> MCP flux, which is generic in
tumor cells [27]; ɸ was considered as a physiological flux of glucose.
Oscillating tumor glycolysis was modeled through the following
Ordinary Differential Equations System (ODES):
d [ F 6P]
= V1 − V2
dt
d [ F1, 6 BP ]
dt
d [ ADP ]
dt
d [ ATP ]
dt
dt
= V2 − V3 − V4
Parameter C was set at 0,0325 mM min-1 a value where the system
exhibits sustained periodic oscillations. It was considered that this value
represents a physiological flux of glucose, based on the conjecture of
Fru et al. [8], who stated that glycolytic oscillations could occur in vivo
tumors. Then a periodic perturbation was applied to the autonomous
model (2.1). Specifically, the glucose influx: C (right member of V1) was
modified by adding the following expression:
=−V1 − V2 + 2V4 + V5 − V8
= V5 − V6 − V7
Table 1. Values of the kinetic parameters used in the glycolytic model
1
2-PFK1
Kinetic parameters
References
C = 0,0325 mM min-1
(a) ɸ
-
k1 = 0,00014 min-1
(a)
-
Vm = 0,04 mM min-1
[3]
n=4
[29,30]
L = 5,6 * 106
(b)
[23]
Ks = 1 mM
(a)
-
Kα = 1 mM
(a)
-
3
k3 = 0,0079 min-1
(a)
-
4
k4 = 0,000154 mM-2 min-1
(a)
-
-1
-1
k5 = 0,001 mM min
5- PYK
(a)
[25]
Kα = 2 mMk
[31]
γ=4
[25]
6- MCP
C1 = 0,0001 mM min-1
(c)
-
7- LDH
8ATPase
k7 = 0,0003 min-1
(c)
-
k8 = 0,0026 min-1
(a)
-
doi: 10.15761/ICST.1000301
A
f
P
( sen 2 ( ft ) + − 1 =
f /r
r
(2.2)
where A and f are the amplitude and the frequency of the perturbation
respectively, t denotes the time of the modeling and r is a constant that
represents the minimum frequency that will be used in the perturbation.
The expression (2.2) is useful to evaluate the effect of periodic
glucose perturbations on the oscillating tumor glycolysis. This
expression only makes sense when f ≥ 1 . Three cases can be analyzed:
r
When f = 1 , the P function takes the particular form used by other
r
authors [34,35]:
Asen2 (fπt) = P
-
z = 0,01
Integr Cancer Sci Therap, 2019
Additionally, the cellular energy charge was estimated from the
ATP/ADP ratio for each value. This index has been previously used
by other authors to estimate cellular energy level [26]. For an oscillatory
dynamic, a mean ATP/ADP ratio was obtained, based on the mean
of the concentrations of ATP and ADP. This value was normalized
in percent with respect to the ATP/ADP ratio corresponding to C =
0,032 mM min-1. For steady state dynamics, the same procedure was
performed, with the exception of the calculation of the mean.
Periodic perturbation of the autonomous model: nonautonomous model
= V1 + V2 − 2V4 − V5 + V8
The initial concentrations of F6P; F1,6BP; ADP; ATP; PEP and Pyr
used in the model were: 1,6 ; 0,5 ; 2,4 ; 14,5 ; 0,5 and 3 mM respectively.
Reaction
In order to characterize the dynamics of the autonomous system
(non-perturbed model) the parameter C was modified in the interval (0
– 0,5) mM min-1 and the stability of the stationary states was determined
using the standard procedure [10]. The time series obtained were
classified according to the dynamic response exhibited.
Therefore, normalization values greater than 100% reflect an
increase in cellular energy charge and lower values, a decrease. The
increase in cellular energy charge activates the metabolic pathways of
biosynthesis, which favors cell proliferation [17].
d [ PEP ]
= 2V4 − V5
dt
d [ Pyr ]
The kinetic model described above constitutes a dynamic
autonomous system because the rate of the reactions does not depend
on time explicitly [32]. C was identified as a control parameter because
it has been proposed as the most important parameter controlling the
oscillations [33] and it can be modified experimentally with increases
or decreases in extracellular glucose concentration.
which can be used to represent the situation in which glucose is
supplied at a rate (Vs) lower than the rate (C) with which the tumor
cell transports it to its interior: Vs < C. As a consequence C oscillates
between the level basal (C = 0,0325 mM min-1) and the maximum rate
reached for that perturbation without “accumulation” occurring. It can
also be used when glucose is deprived at a lower rate than the rate with
which the organism is able to reestablish its levels. These two situations
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Pomuceno-Orduñez JP (2019) Phase transitions in tumor growth VII: The effect of periodic glucose pulses and privations in a cancer model
occur when the frequency of perturbation is much lower than the
autonomous frequency of oscillation of tumor glycolysis.
LZ =
f
When > 1 the P function can be used to simulate frequencies
r
near to the autonomous frequency of oscillation of tumor glycolysis.
Physically, this means that Vs > C . In correspondence, accumulation
occurs and C does not tend to its basal level but oscillates between rates
higher than the basal rate and the maximum rate corresponding to that
perturbation. As the frequency rises, the minimum rate approaches the
maximum rate. It can also be used when glucose is deprived at a higher
rate than the rate at which the organism is able to reestablish its levels.
P = A . In this case, the minimum rate of glucose influx
When lim
f
r
→∞
tends to the maximum rate and a constant glucose influx is established.
This flux is greater than the basal rate of glucose input. Physically, the
constancy in this rate is due to the fact that a balance is reached between
the rate of glucose influx in the tumor cell and the glucose consumption
rate by non-tumor cells in the microenvironment. This prevents the
sustained increase of glucose influx. When glucose is deprived with
an infinite frequency, the constancy in the flux is due to a balance
between the rate of glucose influx in the tumor cell and the rate with
which the organism is able to reestablish its levels. In this case, the flux
is lower than the basal rate of glucose input. This prevents the sustained
decrease of glucose influx.
If the expression (2.2) is incorporated into ODES (2.1) through a
sum, then it represents periodic pulses of glucose. On the other hand,
if it is inserted as a subtraction, then it implies periodic deprivations
of glucose. In both cases it is considered that the external glucose is
consumed by tumor cells and by others that are contained in the tumor
microenvironment.
The values of r used in the simulation of pulses and periodic glucose
deprivations are equivalent to 0,1 f0 and 0,2 f0 respectively, where f0 is
the autonomous frequency of the system (frequency at which species
oscillate in the autonomous system for C = 0,0325 mM min-1.
The dynamic responses of the perturbed system are represented in
a two-dimensional bifurcation diagram, for each pair [f/r; A] imposed.
In this case f/r and A constitute the control parameters. The time series
obtained were classified from a stroboscopic analysis [10]. Additionally,
the cellular energy charge was estimated from the percentage
normalization of the ATP/ADP ratio. For this purpose, the value of the
ATP/ADP ratio of the autonomous system corresponding to C = 0,0325
mM min-1 was taken as reference.
Mathematical procedure
The ODES (2.1) was solved with software COPASI v.4.6 (Build
32) [36]. The numerical method used was Deterministic (LSODA)
with a relative and absolute tolerance of 10-6 and 10-12 respectively. The
variation of the control parameters allowed establishing regions with
different complexity. For each region, the complexity of glycolytic
oscillations was quantified from the LZ index, using the algorithm LZ
- 76 [37]. This index was calculated from the ATP time series because
the dynamics of a species is enough to represent the dynamics of the
whole system [38]. ATP time series was previously coded to a binary
sequence, according to the following criterion: if the ATP concentration
value is greater than the previous one, it was replaced by 0 or 1 if else,
so that the first number in the series was not encoded. The expression
used to calculate the LZ complexity was:
Integr Cancer Sci Therap, 2019
doi: 10.15761/ICST.1000301
p
log 2 N
N
(2.3)
where p is the number of different patterns detected in the binary
sequence and N denotes the number of points in the series.
Since the equation (2.3) is affected by the value of N, the LZ index
was normalized using the equation proposed by [39]:
LZ N =
LZ − LZ NC
LZ NR − LZ NC
(2.4)
where LZNC and LZNR are the LZ values measured for a constant series
(consisting of a single value repeated N times) and random (white
noise) respectively, with the same number of points, N. So LZN is almost
independent of the length of the series [39].
Results
Characterization of the autonomous system
The variation of the control parameter C showed that the sustained
periodic behavior occurs in a range of glucose input rate limited by two
critical values (0,02 and 0,15 mM min-1 approximately). According to
Fru et al. [8] this oscillatory domain could be within the physiological
range. Values of C less than the lower limit or greater than the upper
limit lead the system to a stable steady state.
Table 2 shows the normalized ATP/ADP ratio and the LZ complexity
values associated with each region. In the stable steady state region, LZ
values close to 0 were detected, while higher values were found in the
limit cycle. These dynamic responses were corroborated with a stability
analysis of the stationary states.
The results obtained indicate that the transition from LC to SS2,
due to the sustained increase in glucose influx, can lead to a decrease
in robustness, but can increase cellular energy charge in tumor cells
(Table 2). This transition far from thermodynamic equilibrium is
a consequence of bifurcations. In fact, cancer glycolysis behaves
according to the rules of a ‘‘first order’’ phase transition [15].
Taking into account these results, a metabolic therapy is effective
when it is oriented in two directions: decrease the cellular energy charge
and the robustness of the tumor, or at least prevent the increase of these
two factors. Precisely, the transition from LC to SS1 due to the sustained
decrease in glucose influx, can lead to a decrease in cellular energy
charge and robustness in tumor cells (Table 2). This can be achieved by
directly attacking the degree of vascularization of the tumor.
Table 2. Complexity and energy charge of the autonomous system corresponding to values
of C contained in each of the indicated regions
Normalized ATP/
ADP ratio (%)
Region
C (mM min-1)
LZ
SS1
0
0
0
0.01
0
43
0.0325ɸ
0.003ɸ
100ɸ
0.05
0.004
115
0.08
0.005
117
0.18
0
117
0.5
0
117
LC
SS2
SS1: Stable Steady State obtained for C values lower than the lower limit of the oscillatory
domain; SS2: Stable Steady State obtained for C values greater than the upper limit; LC:
Limit Cycle; ɸ value corresponding to the fixed glucose flux.
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Pomuceno-Orduñez JP (2019) Phase transitions in tumor growth VII: The effect of periodic glucose pulses and privations in a cancer model
Table 3. Complexity and energy charge of the perturbed glycolytic model with periodic
pulses of glucose
Periodic glucose pulses
Figure 2 shows the bifurcation diagram resulting from the
perturbation of the model with periodic pulses of glucose. Three
regions are defined: zone without chemical sense (I), period-k and
quasiperiodicity (II) and period-k (III), where k represents the ratio
between the response period and the period of perturbation.
Region II exhibits dynamics with greater complexity than region
III and the autonomous system (Table 3). Therefore, in region II there
is an increase in the robustness of the tumor. This implies that this zone
should be avoided from a therapeutic point of view. It is notorious that
for small values of A, regardless of the value of f/r, the system exhibits a
highly complex dynamic (Figure 2: región II).
Region
LZ
Normalized ATP/ADP
ratio (%)
Autonomous model (A=0)
0,003 ɸ
100 ɸ
0,02
112
0,02
117
II
III
0,01
110
0,004
117
0,004
116
0,005
113
value corresponding to the fixed glucose flux, obtained when the amplitude of perturbation
A=0. The rest of the values of LZ and the normalized ATP/ADP ratio correspond to pairs
[f/r; A] belonging to each indicated region.
ɸ
It was found that dynamics of period-k are obtained for high
values of A and f/r (Figure 2). These dynamics have a mean complexity
similar to the autonomous model (Table 3). However, when the values
of the normalized ATP/ADP ratio were evaluated for regions II and
III, the values found were higher than the corresponding ones to the
autonomous model (Table 3). This could mean that an increase in the
proliferative capacity of the tumor cells may occur. These results showed
that it is not convenient to use periodic pulses as therapy in tumors that
exhibit oscillating glycolysis.
Table 4. Complexity and energy charge of the perturbed glycolytic model with periodic
glucose deprivation
Periodic glucose deprivation
value corresponding to the fixed glucose flux, obtained when the amplitude of the
perturbation A=0. The rest of the values of and the normalized ATP/ADP ratio correspond
to pairs [f/r; A] belonging to each indicated region.
Figure 3 shows the bifurcation diagram resulting from the
perturbation of the model with periodic glucose deprivation. The
qualitative behaviors exhibited are the same as those achieved with
periodic pulses: period-k and quasiperiodicity (region I) and period-k
(region II). The fundamental difference between the dynamic responses
obtained by both forms of perturbation lies in the area that occupies
the most complex region (darker zone) in the bifurcation diagrams.
This area becomes smaller when the model is forced with periodic
deprivation.
The robustness of the tumor only increases in a narrow domain
of and f/r, bounded by small values of f/r (Figure 3: darker zone).
Higher values of f/r led the system to a state where its mean complexity
is similar to the autonomous model (Table 4: region II). It is notorious
that the value of the normalized ATP/ADP ratio decreased in both
regions with respect to the autonomous model (Table 4). Therefore, it
would be convenient to periodically deprive the tumor cells of glucose
using high values of perturbation frequency, since it could prevent the
increase of the resistance capacity of the tumor cells and contribute to
Region
LZ
Normalized ATP/ADP
ratio (%)
Autonomous model
(A=0)
0,003 ɸ
100 ɸ
I
0,009
57
0,009
66
0,006
71
0,002
3
II
ɸ
Figure 3. Bifurcation diagram for the perturbed glycolytic model with periodic glucose
deprivation
A: Amplitude of perturbation; f/r: ratio between the frequency of perturbation and the
minimum frequency used in the perturbation. Region I: period-k and quasiperiodicity;
Region II: period-k. The dynamic responses were classified by stroboscopic analysis.
the decrease of its proliferative capacity. This strategy can increase the
efficacy of antitumor therapies.
Discussion
When the glucose input rate was varied, the updated model
(autonomous model) simulated an oscillatory domain. This coincides
with experimental findings reported in the literature [33,40]. Therefore,
the modifications made to the previous model [15] were effective.
Reliability in predictions made from periodic variations of the glucose
input rate was guaranteed due to the existence of this oscillatory
domain.
Figure 2. Bifurcation diagram for the perturbed glycolytic model with periodic pulses of
glucose
A: Amplitude of Perturbation, f/r: Ratio between the frequency of perturbation and the
minimum frequency used in the perturbation. Region I : Zone without chemical sense
(negative concentrations of the intermediaries); Region II: period-k and quasiperiodicity;
Region III: Period-k. The dynamic responses were classified from stroboscopic analysis
Integr Cancer Sci Therap, 2019
doi: 10.15761/ICST.1000301
The molecular mechanism of autonomous oscillations in glycolysis
was proposed by Goldbeter [33], according to the theory developed by
Monod, Wyman and Changeux [24]. The mechanism proposes that
high rates lead to a predominant relaxed state (R) in the enzyme PFK-1;
while low rates lead to a predominant tense state (T). The predominance
of T or R leads to a stable steady state; while the transitions (T-R) lead to
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Pomuceno-Orduñez JP (2019) Phase transitions in tumor growth VII: The effect of periodic glucose pulses and privations in a cancer model
sustained oscillations. These transitions occur at intermediate substrate
input rates.
The application of periodic pulses of glucose caused quasiperiodic
and period-k dynamics. These dynamics were found experimentally
in cell-free extracts obtained from yeast that exhibited autonomous
oscillations in the glycolytic mechanism, which were perturbed with
periodic substrate input flux [20,21].
The results obtained show that the application of periodic pulses of
glucose should be avoided from a therapeutic point of view. In contrast,
a study conducted on human endothelial cells revealed that prolonged
exposure to periodic pulses of external glucose leads to oxidative stress
and cellular DNA damage. This can activate p53 tumor suppressor
protein and trigger apoptosis [41]. However this is not a generality in
tumors since there are allelic variants of the gene that encodes p53 that
decrease the sensitivity of the tumor cell to apoptotic signals [42].
Predictions made from the application of periodic glucose
deprivation agree with the results reported experimentally by Lee et
al. [22], which showed that nutrient deprivation cycles affected the
energy metabolism of different tumor cell lines, which was manifested
through the arrest of the proliferative capacity of them. In vivo studies
have shown that low blood glucose concentrations can increase the
cytotoxicity of chemotherapeutic compounds [22,43]. This supports
the possible effectiveness of the proposed strategy.
Mutations and epigenetic modifications that increase growth and
promote insensitivity to anti-growth signals in cancer cells, lead to the
loss of appropriate responses to rapidly adapt to a variety of extreme
environments including starvation [22]. Recently a study based on a
kinetic model of tumor glycolysis adjusted to steady state, showed
that the application of intermittent fasting decreased the entropy
production rate of glycolysis. In this study, the entropy production rate
is considered as an indicator of robustness [44].
An important aspect to keep in mind is that therapies aimed at
damaging glucose metabolism may fail due to the heterogeneity and
the evolutive capacity of tumor cells [45]. It has been shown that some
cancer cells have acquired a strong tolerance to stress [46], like the
one induced by nutrient deprivation. Therefore, in order to improve
the effectiveness of this strategy to a wide variety of tumors, it is
recommended avoid the stress tolerance. The use of certain drugs has
given good results in this regard [46,47].
Conclusion
Our results show how the environmental constraints, represented
in this case by the availability of glucose, can affect the robustness and
energy charge of neoplastic cells that exhibit oscillating glycolysis. The
tumor microenvironment plays an important role in the fate of tumor
cells [48]. Therefore, the design of chronotherapies that adequately alter
the cancer environment can promote current therapies against it.
In summary, in this study we found:
Sustained increases in glucose influx causes a decrease in the
robustness, but causes an increase in the cellular energy charge of
tumor cells that exhibit an oscillating glycolysis.
perturbation, prevent the increase of robustness and cause a decrease
in the cellular energy charge. This strategy can increase the efficacy of
antitumor therapies.
Acknowledgments
Prof. Dr. A. Alzola in memoriam. We would like to thank Prof. Dr.
Jacques Rieumont for support and encouragement for this research.
One of the authors (JMNV) thanked the Institute of Physics of the
UNAM Mexico for its warm hospitality.
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Copyright: ©2019 Pomuceno-Orduñez JP. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Integr Cancer Sci Therap, 2019
doi: 10.15761/ICST.1000301
Volume 6: 7-7