Kinetic Modeling of the Blood Oxygenation Level Dependent (BOLD) Signals and Biocatalytic Reactions Observed in the Human Brain Using MRI: An Analysis of Normal and Pathological Conditions

Article abstract of DOI:10.1021/acschemneuro.9b00671

A kinetic model describing the pulse of increased oxygen concentrations and the subsequent changes in the concentration of N-acetylaspartate in the excited nervous tissue of the human brain in response to an external signal is presented. The model is based on biochemical data, a multistage and nonlinear dynamic process the BOLD signal and N-acetylaspartate. The existence of multiple steady states explains the triggering effect of the system. The inhibitory effect of the substrate is a necessary factor for the autostabilization of N-acetylaspartate. The kinetic model allows the dynamic behavior of previously unmeasurable metabolites, namely, products of the hydrolysis of N-acetylaspartate, such as acetic and aspartic acid, and glutamic acid to be predicted. Kinetic modeling of the BOLD signal and the subsequent hydrolysis of N-acetylaspartate provides information about the biochemical and dynamic characteristics of some pathological conditions (schizophrenia, Canavan disease, and the superexcitation of the neural network).

Full text of DOI:10.1021/acschemneuro.9b00671

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Article
Kinetic modeling of the BOLD signals and biocatalytic
reactions observed in the human brain using MRI:
an analysis of normal and pathological conditions
Sergey D. Varfolomeev, Valeriy I. Bykov, Nataliya N. Semenova, and Svetlana B. Tsybenova
ACS Chem. Neurosci., Just Accepted Manuscript • DOI: 10.1021/acschemneuro.9b00671 • Publication Date (Web): 10 Feb 2020
Downloaded from pubs.acs.org on February 23, 2020
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KINETIC MODELING OF THE BOLD SIGNALS
AND BIOCATALYTIC REACTIONS OBSERVED
IN THE HUMAN BRAIN USING MRI: AN
ANALYSIS OF NORMAL AND
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PATHOLOGICAL CONDITIONS
Sergey D. Varfolomeev,
Lomonosov Moscow State University, Institute of Physical and Chemical Grounds of
Neuronet Functions and Artificial Intelligence, Moscow, 119991, Russia,
Emanuel Institute of Biochemical Physics, Moscow, 119334, Russia,
Valeriy I. Bykov,
Emanuel Institute of Biochemical Physics, Moscow, 119334, Russia,
Nataliya A. Semenova,
Emanuel Institute of Biochemical Physics, Moscow, 119334, Russia,
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Svetlana B. Tsybenova*
Emanuel Institute of Biochemical Physics, Moscow, 119334, Russia, +79153049387,
s.tsybenova@gmail.com
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Keywords: fMRI, BOLD signal, N-acetylaspartate, kinetic model, dynamic process,
schizophrenia
ABSTRACT
A kinetic model describing the pulse of increased oxygen concentrations and the
subsequent changes in the concentration of N-acetylaspartate in the excited nervous
tissue of the human brain in response to an external signal is presented. The model is
based on biochemical data, a multistage and nonlinear dynamic process the BOLD
signal and N-acetylaspartate. The existence of multiple steady states explains the
triggering effect of the system. The inhibitory effect of the substrate is a necessary
factor for the autostabilization of N-acetylaspartate. The kinetic model allows the
dynamic behavior of previously immeasurable metabolites, namely, products of the
hydrolysis of N-acetylaspartate such as acetic and aspartic acid, and glutamic acid to be
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predicted. Kinetic modeling of the BOLD signal and the subsequent hydrolysis of N-
acetylaspartate provides information about the biochemical and dynamic characteristics
of some pathological conditions (schizophrenia, Canavan disease and the
superexcitation of the neural network).
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INTRODUCTION
An understanding of physical and chemical processes of the nervous system that are
activated in response to external signals, as well as the receipt, storage and processing
of information is one of the most attractive and challenging tasks of modern science. In
recent decades, improvements in methodological approaches have increased our
understanding of the mechanisms underlying these processes. Noninvasive methods
such as functional magnetic resonance imaging (fMRI) and NMR spectroscopy allow
researchers to observe the neurophysiological and chemical processes in the human
brain and animals in vivo [1–6].
These methods are widely used to study processes in the human brain and provide
morphological descriptions and diagnoses for a number of pathologies, including
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traumatic brain injury, schizophrenia, Canavan disease, Alzheimer’s disease, and
Parkinson’s disease [7–16].
A key process in the cascade of biochemical and physiological reactions associated
with the excitation of neural networks in response to external stimulus is the BOLD
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(blood oxygen level dependent) signal, namely, the impulsive enrichment of a zone of
excitation in the neural network by oxygenated blood [17–22]. One of the key stages of
the function of neural networks is an increase in the oxygen supply of the activated
regions of the neural network (hemodynamic response). The BOLD signal enables MRI
to visualize the zones of excitation. Obviously, this process is initiated by the induced
expansion of microvessels (microvasodilatory effect). The chemical nature of this initial
step is one of the fundamental problems that require solutions. The microvasodilation
process, which is induced by thought processes and the excitation of local neural
networks, has a pulsed character. The goal of this study is to create a representation of
the biocatalytic nature of chemical reactions underlying the BOLD signal and a kinetic
model of the process.
Importantly, NMR spectroscopy allows researchers to directly identify a number of key
metabolites and neurotransmitters present in the human brain and creates the
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possibilities of studying dynamic processes involving these molecules [23–26].
Descriptions of the kinetics of these processes, mathematical modeling of the dynamics
and a study of the dynamic behavior of components based on the assumption of a
possible chemical scheme of biocatalytic transformation of substances is a key stage in
understanding the function of the brain.
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One of the most important molecules in the brain is N-acetyl-aspartic acid (N-
acetylaspartate). N-acetylaspartate is a biomarker of neurons in the brain [27, 28]. This
compound is almost exclusively detected in the nervous system and is absent in other
human and animal organs and tissues. Its concentration in the mammalian brain is
approximately almost 10 µmol/g, making it the second most common derivative of free
amino acids in the central nervous system after glutamate. Because of the high
concentration, N-acetylaspartate levels are quantitatively determined using NMR
spectroscopy in vivo [12, 13]. The role of N-acetylaspartate in brain metabolism has not
been completely clarified. It is synthesized by the acetylation of aspartic acid. The
hydrolysis of this compound by N-acetylaspartate hydrolase (aspartoacylase) is a key
metabolic reaction.
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An experimental study of the dynamic changes in the concentration of N-
acetylaspartate detected a temporary decrease in its concentration with the yield at the
initial level (the “anti-peak”) [29]. A model of the process considering the nonlinearity of
the inhibition of aspartoacylase by excess substrate has led to the conclusion of the
trigger nature of the process and kinetic effects of autostabilization of the concentration
of this key metabolite [30, 31].
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Thus, the BOLD wave initiates the hydrolysis of N-acetylaspartic acid, and both of
these processes are sufficiently reliably detected in vivo in the human brain using fMRI
and NMR spectroscopy. Kinetic modeling can predict the activation of several
secondary processes, namely, the formation of products (aspartic acid and acetic acid)
and potential mediator of excitation glutamic acid.
Changes in dynamic characteristics of the BOLD signal are manifested in the neural
networks of patients with a number of pathological conditions. Schizophrenia is
characterized by a decrease in the maximum values of the BOLD signal [24]. On the
other hand, a common genetically transmitted disease of the brain, Canavan disease, is
associated with the hyperproduction of N-acetylaspartic acid due to the low catalytic
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activity of aspartoacylase. Modeling helps explain the molecular mechanism of these
deviations at the kinetic level.
Thus, the tasks that were performed in the present study are listed below.
1) Based on data from literature, a biochemical sequence of reactions describing the
processes included in the implementation of the BOLD signal was created and a
quantitative description of the experimentally observed dynamics of this phenomenon is
provided.
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2) A mathematical description of the subsequent effect of the BOLD signal on the
kinetic process of the hydrolysis of N-acetylaspartic acid (dynamics of “anti-peak”) was
created and compared with theoretical and experimental data.
3) Predictions of the dynamic generation of products of the hydrolysis of N-
acetylaspartic acid, namely, aspartate and acetate, and subsequent production of
glutamic acid were created.
4) The possible effects of the system factors identified in the biochemical study on the
dynamic behavior of the brain in patients with various neuropathologies, namely,
schizophrenia, Canavan disease, and potential overexcitation of the neural network,
were analyzed.
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RESULTS AND DISCUSSION
Kinetic Model of the BOLD signal. Previously, a series of experiments aimed at
identifying of a number of chemical processes that activated in the system of neurons
responsible for sensory perception was performed [23–26]. Sensory (visual and sound)
signals and the necessity of solving the problem stimulate different areas of the cortex
and other parts of the brain. In fMRI, these changes are manifested in the images of
different areas of the brain obtained using superparamagnetic oxygenating hemoglobin
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[
17–19, 32, 33].
Fig. 1 (line with dots) shows a typical dynamic pattern of the BOLD signal [18]. The
sensory perception of a signal, neural network information processing including,
memory elements, and determining the subsequent response are sufficiently fast
processes. Their time range is less than a second. A local increase in the concentration
of oxygenated hemoglobin (BOLD signal) is a general phenomenon observed in
response to an external stimulus. As shown in our previous study, a pulsed increase in
the oxygen concentration is followed by a series of delayed chemical processes, in
particular, the hydrolysis of N-acetylaspartic acid [23, 24].
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Several difficulties occur when attempting to answer the question “How is the
excitation of neurons following stimulation with a visual or sound signal (generation of
potential and ion current) converted into vasodilatory function (extension of
microvessels and an increase in the oxygen flow)?”. The pairing of these processes in a
sequence of biocatalytic stages requires a common intermediary.
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Figure 1. Experimental [18] and calculated curves of the dynamic changes in oxygen
concentrations in response to an external signal by varying the parameter k : 1 –
B
–
1
–1
–1
k = 0.31 s , 2 – k = 0.15 s , and 3 – k = 0.05 s .
B
B
B
The regulated behavior of vessels, including microvessels, in the brain is consistent
with established representations of the effects of the classical vasodilators nitric oxide
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NO and/or some prostaglandins (PG), in particular prostacyclin (PGI ) or prostaglandin
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E , on the mechanical behavior of endothelial cells covering the vessel walls [32, 34].
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A possible biochemical mechanism underlying the process is described below.
Calcium ions represent a general facilitator of the flow of ions and behavior of the
microvessels in the brain following the excitation of neurons. An increase in permeability
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to Ca cations occurs upon neuronal depolarization [34]. This increase in permeability
2+
is required for muscle contraction [20]. Ca ions are the main regulator of the synthesis
2+
of vasodilators. In the case of neuronal NO synthase, Ca ions, being incorporated into
the enzyme structure, activate this enzyme and “close” reductase and oxygenase
domains, inducing the synthesis of NO from L-arginine [35, 36]. Regarding the potential
involvement of prostaglandin vasodilators in these processes, Ca2+ ions are specific
activators of phospholipase A , which is isolated from the phospholipid arachidonic acid.
2
Arachidonic acid is the basic substrate for the synthesis of prostaglandins [34].
In the present, a kinetic model of changes in the oxygen concentration occuring as a
result of a sequence of biocatalytic transformations in neuronal networks of a healthy
person and patients with some neuropathological conditions is proposed and solved.
The kinetic scheme of the process is represented as follows:
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kn
k
k1
kB
V
A1
A2...n
B
V
R
kB
k
V
k
R
B
V
R
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(1)
2+
where A is the first stimulus (Ca ); A2...n, B are intermediate compounds; V are
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vasodilators (potentially NO and PG), and R is the difference in oxygen concentration
(
[O ]), which is experimentally measured before and after the external stimulus. We
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postulated that R depends linearly on the concentration of the vasodilator V. The
corresponding processes for removal (output) of the variables B, V and R from the
system are described by the rate constants k , k , and k , respectively:
B
V
R
d[A1]


k [A ],
1
1
dt
d[A2 ]
k [A ]  k [A ],
1
1
2
2
dt

         
d[A5 ]

k [A ]  k [A ],
(2)
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4
5
5
dt
d[B]
dt
d[V ]
dt
 k [A ]  (k  k )[B],
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5
B
B


k [B]  (k  k )[V ],
B
V
V 
d[R]
dt
k [V ]  k [R],
V
R
where k are rate constants of stages of scheme (1) and [A ],..., [A ], [B], [V], and [R] are
i
1
5
the concentrations of substances A ,..., A , B, V, and R respectively.
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For the kinetic model (2), the initial conditions are t  0 : [A ](0)  A (pulse initiator),
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1
[A ](0)  [B](0)  [V ](0)  [R](0)  0 , i = 2,..., 5.
i
The solution of system (2) (Fig. 1, curve 1) describing the dynamics of the BOLD
signal was compared with the experimental curve presented in a previous study [1]. The
calculations were performed with the following values for the kinetic parameters and
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
–
1
–1
–1
–1
initial conditions: k = k = k =k = k =1 s , k = 1.2 s , k = 0.31 s , k = 2.6 s ,
1
2
3
4
B
5
B
V
–
1
–1
k = 0.8 s , k = 18 s , and [A ](0) = 1. The dependence of the concentration [R](t)
V
R
1
(curve 1 is shown by the line with dots in Fig. 1) in model (2) is consistent with the
experimental data presented in previous study [18].
The kinetic model based on biochemical representations of the pairing of a nerve
pulse with the synthesis of a neurovasodilator (V) following an increase in the oxygen
level subject to certain kinetic parameters of a system can predict the kinetic behavior of
variables, that are unable to be directly determined experimentally, namely, A, B, and
i
V. The number of stages n in the chain of transformations determines the extent of the
delay in the BOLD signal: at more intermediate stages, the delay time of the system
response to an external signal increases. It can be shown that the dynamics of the sequential
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scheme of transformations under some simplifying assumptions is well described by the
dependence ΔO (t) [31]:
2
n
t

O (t)  t e ,
(3)
2
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
5
5
5
5
5
5
5
5
5
5
6
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
where α is a parameter determined from the extreme point of the ΔO (t) dependence; n is the
2
number of stages in sequence (1); β is the average value of the rate constants of reactions in the
chain of transformations. It is easy to show that the maximum of dependence (3) after
differentiation is defined as:
n
n
tmax  n/ , O (t ) tmax e .
(4)
2
max
Simple equations (4) make it possible to estimate the parameters in model (2) from experimental
data. Comparison of dependence (3) with the experimental data gives the estimates n ≈
–1
5
and β ≈ 1 s . Thus, there are about five stages, which are the kinetically important
stages preceding the vasodilator process. The absence of stages A A2...nB... leads
1
to the disappearance of the initial portion of the delay in the BOLD signal.
Kinetic description of an “anti-peak” of n-acetylaspartate hydrolysis. Trigger effect and
autostabilization. The kinetic behavior of N-acetylaspartate (S) consists of a few
processes.
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1
1
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1
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2
2
2
2
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3
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6
–
The system is open on the substance S; a zone of excitation is induced by the
substance following an increase in blood flow and its transport from the zone of
excitation.
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
– The system includes the biochemical transformation of N-acetylaspartate by
aspartoacylase to produce acetic and aspartic acids.
– Since the occurrence of the kinetic “anti-peak” of N-acetylaspartate is a response to
the occurrence of excess oxygen in the excitation zone, the oxidative process should be
included in the description of the dynamic changes in the N-acetylaspartate
concentration. In biochemistry, the processes responsible for the direct oxidation of N-
acetylaspartate are unknown. Oxidation proceeds through several intermediate stages,
such as the tricarboxylic acid cycle with subsequent conversion of the reductive
equivalents through a chain of electron transfer in the mitochondria.
The observed kinetic responses of N-acetylaspartate to the excitation of the neural
network are shown in Fig. 2. The BOLD response corresponds to a change in the
concentration of oxygenated hemoglobin in the studied volume of neuronal tissue. After
a delay (induction period), the oxygen concentration increases, peaks at 6 s and then
returns to the starting basal level (see also Fig. 1). The change in the N-acetylaspartate
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concentration occurs after a 5–6 s delay, while the “anti-peak” is observed – a
temporary reduction in the N-acetylaspartate concentration. The minimum value is
observed at the 12th second after the touch stimulus.
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
3
3
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4
4
4
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4
4
4
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5
5
5
5
5
5
5
5
5
6
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
The intermediate stages, reflecting the possibility of oxygen participation before
interaction with the N-acetylaspartate (S), are added to the scheme of reactions (1):
k6
1
) R  AA  AA ,
1
2
k7
2
) AA  AA ,
(5)
2
3
k8
3
) AA  S  AA S,
3
3
where AA , AA , and AA are the intermediate substances involved in the oxidation
1
2
3
process (for example, AA is one of the key metabolites of energy metabolism, included
1
in the tricarboxylic acid cycle or in process of oxidative phosphorylation).
Corresponding to the scheme of reactions (1) and (5) a general kinetic model of the
dynamic changes in the BOLD signal ([R]) and the N-acetylaspartate concentration ([S])
in the system “external stimulus – response of the brain” is provided by the following
equations:
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1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
5
5
5
5
5
5
5
5
5
5
6
d[R]

k [V ]  k [R]  k [R][AA ] ,
V
R
6
1 0
dt
d[AA2 ]

k [R][AA ]  k [AA ],
6
1
0
7
2
dt
d[AA3]
(
6)

k [AA ]  k [AA ][S],
7
2
8
3
dt
d[S]
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0

(S [S])  v([S])  k [AA ][S],
0
8
3
dt
where k represents the rate constant of the intermediate stages (5). The term α(S –
i
0
[
S]) reflects the open nature of the system. Substance S is included into the reaction
volume of the S concentration and after enzymatic transformation emerges from this
0
volume the concentration [S]. v([S]) is the hydrolysis rate of S mediated by
aspartoacylase, and k [AA ][S] is the rate of the oxidative process involving one of the
8
3
metabolites of S after conversion. For simplicity, without prejudice regarding the
generality of the approach, the component AA is assumed to be present in a large
1
excess and its concentration is constant.
The reaction of hydrolysis of N-acetylaspartate has been quite well studied [37]. A
characteristic feature of the regulation of enzymatic activity, which certainly plays an
important role, is a complex dependence of the reaction rate on the concentration of the
substrate.
3
V [S]
m
v([S]) 
.
(7)
2
3
4
K  K [S]  K [S]  K [S]  K [S]
m
1
2
3
4
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A molecular-kinetic study of these effects based on QM/MM approximations is
provided in previous studies [38–41].
At high concentrations of substrate, the reaction rate is significantly reduced (inhibition
by excess substrate). Importantly, to emphasize that the physiologically observed N-
acetylaspartate concentration in neurons will significantly suppress an active enzyme
(K , K , K , K , and K ).
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
4
4
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4
4
4
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4
4
4
5
5
5
5
5
5
5
5
5
5
6
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
m
1
2
3
4
The proposed kinetic model quite adequately describes the experimental data (Fig. 2),
including the “delays” occurring during the kinetic stages of accumulation of products
and the “anti-peak” of the N-acetylaspartate concentration. The low accuracy of the
experiment does not allow us to determine the values of process parameters. We were
only able to estimate the ranges in which “delays” quite accurately are described by the
model.
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0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
Figure 2. Experimental and calculated curves showing changes in the O concentration
2
([R]) and N-acetylaspartate concentration ([S]) after exposure to an external signal: the
dotted lines represent experimental data (the BOLD signal and N-acetylaspartate
concentration) and solid lines represent calculated lines ([R] and [S]). The results of the
experiment and simulation were normalized to the amplitude of the N-acetylaspartate
concentration at time t = 0.
The curves shown in Fig. 2 are the solutions of the system of equations (2), (6) and
–1
(
7) with the following values for the parameters: α = 6,  = 1, S = 0.008, k = 1 s
0
i
–
1
–1
–1
–1
–1
–1
(
i =1,...,4), k = 0.4 s , k = 8.8 s , k = 0.3 s , k = 65 s , k = 0.045 s , k =1 s ,
5
6
7
8
B
B
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–
1
–1
–1
–13
–9
k = 2.6 s , k = 0.8 s , k = 20 s , V = 0.248, K = 1.810 ; K = 910 ,
V
V
R
m
m
1
–
4
3
K = 4.510 , K = 2, K = 2.4410 , [AA ] = 1, [A ](0) = 1, [A](0) = 0 (i = 2,...,5),
2
3
4
1 0
1
i
[
AA ](0) = [AA ](0) = 0, and [S](0) = S . The values of V , K , K , K , K , and K were
2
3
st
m
m
1
2
3
4
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
5
5
5
5
5
5
5
5
5
5
6
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
determined from experimental data as described in previous studies [37–41]. Notably,
the maximum oxygen concentration (for control group) was observed at the 6th second
after exposure to the external stimulus and the minimum N-acetylaspartate
concentration was observed at the 12th second. Within 30–33 s, the concentration of N-
acetylaspartate generally reached its original steady level.
Using an adequate kinetic model, the following key questions that provide a
representation of the behavior of the “mysterious” N-acetylaspartate system can be
answered:
–
What is the mechanism of autostabilization of the N-acetylaspartate concentration
as a key metabolite in the brain? What is the role of the inhibition of the activity of
aspartoacylase by excess substrate this process?
–
What are the dynamic responses in the behavior of the subsequent chemicals in a
chain of transformation of metabolites (aspartic acid, acetic acid, and glutamic acid)
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5
6
after “a wave” of hydrolysis of N-acetylaspartate and the formation of the appropriate
products?
The inhibition of an enzyme by excess substrate in an open system following the
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
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9
0
1
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9
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1
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9
0
1
2
3
4
5
6
7
8
9
0
introduction of substance S with an initial concentration S and the output from the
0
system of [S] leads to the formation of only 1 or 3 steady states [42, 43]. The several
steady states occur due to the non-linear dependence in (5).
A graphical solution for the equation describing the stationary kinetics of the
enzymatic reaction in an open system (the term α(S – [S])) under conditions of
0
inhibition by excess substrate is shown in Fig. 3. When α ranges from 5 to 8, the
presence of three steady states is a characteristic of the system. In this case, steady
states 1 and 3 are stable, but steady state 2 is unstable.
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Figure 3. Dependence of the reaction rate (v) on the substrate concentration ([S]) by
varying α (α is the parameter of the rate input and output of the substrate) in the system
equations (2), (6) and (7) in the presence and absence of substrate inhibition. Points 1,
2 and 3 are the steady states of the system.
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
4
4
4
4
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4
4
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4
4
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5
5
5
5
5
5
5
6
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
The system is quite sensitive to the initial concentration [S](0) (see Fig. 4). In the
presence of high [S](0) (curves 1–4), the system tends to adopt stable steady state with
low catalytic activity of aspartoacylase and high [S] , respectively. At the low initial
st
concentrations [S](0) (curves 5 and 6), a high catalytic activity and virtually no N-
acetylaspartate are characteristic for the enzyme. The possibility of the existence of
three steady states (two stable and one unstable) explains the trigger effect of the
system: small changes in the initial conditions (curves 4 and 5) produce significant
changes in the steady state of the system. Importantly, the system is exceptionally
sensitive to the initial conditions [S](0): the transition from high [S] (curve 4) to low
st
(
curve 5) is implemented at changing [S](0) 5.510–3 %. In the absence of substrate
inhibition (Fig. 3, dashed line), the system is unable to reach the original steady level of
the N-acetylaspartate concentration.
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1
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9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
Figure 4. Dynamic changes in the substrate concentration over time under different
initial conditions [S](0) of the system using equations (2), (6) and (7) (“trigger effect”).
Fig. 5 illustrates the trigger characteristics of the system behavior. The area of the
“
trigger effect” depends on values such as S , α, V , and K . As shown in Fig. 5, the
0
m
4
area of the “trigger effect” (the shaded area) is expanded and moved to the right as the
catalytic activity of the enzyme V increases. At values of [S] > 0.02, the nature of
m
st
change of [S] (S ) becomes almost linear. Therefore, in the presence of large initial
st
0
values of S , the N-acetylaspartate concentration also increases.
0
A fundamental conclusion of this study is that the effect of substrate inhibition on this
system is a necessary factor contributing to the autostabilization of the N-
acetylaspartate concentration as a key metabolite in the brain.
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0
1
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1
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0
1
2
3
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9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
Figure 5. Dependence of the stationary substrate concentrations ([S] ) on the
st
concentrations of input S with the following values for the parameter V : 1 – 0.05, 2 –
0
m
0.248, and 3 – 0.6.
Secondary wave. In the biochemical scheme, the hydrolysis of N-acetylaspartate and
formation of aspartic and acetic acids induces a chain of subsequent reactions leading
to the generation of key metabolites in the brain.
N-acetylaspartate is a metabolite that plays a key role in two important systems of
neuronal metabolism: energetic (formation of acetate) and neuromediatory (formation of
aspartic acid) metabolism. Active metabolites involved in neurobiological processes and
energy metabolism are the products of N-acetylaspartate hydrolysis. Acetate produced
from N–acetyl–CoA during the synthesis of the neurotransmitter of acetylcholine
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participates in anaerobic energy processes (Krebs cycle). Aspartic acid is a
neuromediator, but its most important role occurs after its transformation into glutamic
acid as the main neurotransmitter used in a significant proportion of neurons [42–46].
Glutamic acid is produced from rapid process of reamination using α-ketoglutaric acid,
and the subsequent conversion of glutamic acid produces γ-aminobutyric acid, an
essential inhibitory neurotransmitter. Enzymatic hydrolysis of N-acetylaspartate leads to
the formation of two products: aspartic acid and acetic acid. Therefore, the kinetic
equations, describing the hydrolysis of formation aspartic acid and acetic acid, are
similar. Accordingly, the dynamics behavior of acetic acid is very close to the dynamics
of formation and consumption of aspartic acid. A secondary wave of acetic acid is close
to the secondary wave of aspartic acid.
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The formation of aspartic acid (Asp) and acetate are strictly stoichiometric processes
occurring during N-acetylaspartate consumption. In the dynamic model (2), (6) and (7),
the rate of consumption (–v([S])) changes the signs to positive:
d[Asp]

(S [S])  v([S])  [Asp],
(8)
0
dt
where [Asp] is the transformation of aspartic acid into glutamic acid:
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d[Glu]

 [Asp] [Glu],
(9)
dt
and [Glu] is the rate of Glu transformation in subsequent metabolic pathways.
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The results of numerical integration of the equations (2) and (6)–(9) are shown in Fig.
6
with =10 and =0.3.
Figure 6. Theoretical and experimental curves showing changes in the concentrations of
aspartic ([Asp]) and glutamic ([Glu]) acids, oxygen level ([R]) and N-acetylaspartate ([S])
]
) in accordance with the solutions of the equations (2) and (6)–(9). Solid lines show the
–1
theoretical changes in the concentrations in the normal patients (k = 0.045 s , k = 8.8
B
6
–
1
–1
–1
s , k = 0.3 s , and k = 65 s ). Dashed lines depict the dynamic waves in a group of
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–1
–1
–1
–1
patients with schizophrenia (k = 0.02 s , k = 4.4 s , k = 0.1 s , and k = 195 s ).
B
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Importantly, the autostabilization of the [Asp] and [Glu] concentrations is determined
by the substrate mediated inhibition of aspartoacylase.
Steady state at the high activity of N-acetylaspartate hydrolysis and high levels of the
excitatory signal. Bifurcation transition to “superexcitation”. Kinetic modeling of the
BOLD signal and the coupled hydrolysis of N-acetylaspartate predicts the existence of
two steady states that are characterized by the following features:
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1. A high stationary level of N-acetylaspartate concentration and a low enzyme activity
because of the substrate mediated inhibition (Steady state I, Fig. 3), and
2. A high enzyme activity and a low stationary concentration of N-acetylaspartate
(Steady state II, Fig. 3).
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Figure 7. Bifurcation transition of the [S] concentration at varying the activity of the
enzyme Vm.
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Figure 8. Bifurcation transition of the [S] concentration on time at changing the rate of
synthesis of vasodilator (kB).
The transition of the system in steady state II is determined by the high level of activity
of aspartoacylase (Fig. 7) or the high rate of synthesis of the vasodilator (Fig. 8). Fig. 7
presents the dynamic response of the system as the maximum activity of
aspartoacylase (V ) increases. As shown in Fig. 8, an increase in the rate of synthesis
m
of the vasodilator (parameter k ) causes the system to shift (bifurcation transition) to a
B
stationary state II characterized by the substantial hydrolysis of N-acetylaspartate.
Physiologically, this mode is implemented under conditions of “superexcitation” (Fig. 7)
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through the following mechanism: a powerful touch stimulus  induction of the release
of a large amount of Ca2+  the synthesis of a vasodilator  a significant increase in
the oxygen concentration  a significant reduction in [S]  a significant decrease in the
release of aspartic, acetic and glutamic acids.
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Dynamics of the bold signal and the production of a wave of metabolites in patients
with schizophrenia. In patients with schizophrenia, a maximum level of the BOLD signal
was reduced [10, 15, 16]. Patients with the early stages of schizophrenia (a group with
pathology) present a peak of the BOLD signal that is 2 times less than the peak of the
BOLD signal detected in the control group [26]. The kinetic model ((2) and (4)–(7)) quite
adequately reflects these differences (see Fig. 6). As shown in Fig. 6, the concentration
of N-acetylaspartate in the group with the pathology is almost constant, and no clearly
defined “anti-peak” of hydrolysis of N-acetylaspartate is observed. The calculations for
patients with schizophrenia were performed by changing the values of the parameters
–
1
–1
–1
–1
k = 0.02 s , k = 4.4 s , k = 0.1 s , and k = 195 s .
B
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In the presence of a decreasing constant k (the rate of synthesis of vasodilator V),
B
the BOLD signal decreases significantly (see Fig. 1). This finding is consistent with the
data presented in a previous study [9] showing that the amplitude of the BOLD signal
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triggered by an external stimulus is significantly reduced in the area of primary motor
cortex in patients with the early stages of schizophrenia. In the framework of the
developed model, this change should lead to a significant reduction in the value of the
“anti-peak”, namely, a significant decrease in the wave of N-acetylaspartate hydrolysis
and a subsequent reduction in the production of its hydrolysis products (acetic and
aspartic acids). This result was confirmed experimentally. The data presented in a
previous study [24] and results of calculation model (2) and (4)–(7) are presented in Fig.
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6. As shown in Fig. 6, patients with schizophrenia almost completely lack the wave of N-
acetylaspartate hydrolysis. This difference may be one of the causes or consequences
of this pathology.
Canavan disease. Canavan disease is associated with kinetic features of
aspartoacylase behavior. This disease, which was described in the 1930s of the last
century by M. M. Canavan, is inherited genetically and is common in the Ashkenazi
Jewish population. According to statistics, 1 in 40 persons of Ashkenazi Jewish heritage
is affected by this disease [12–14]. The disease manifests as mental retardation and
destruction of the myelin sheaths of neurons [37]. Biochemically, the disease is
characterized and diagnosed by the hyperproduction of N-acetylaspartic acid in the
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