=Paper=
{{Paper
|id=Vol-1638/Paper86
|storemode=property
|title=Modeling of ambient glutamate concentration measurement in the mammalian nervous system
|pdfUrl=https://ceur-ws.org/Vol-1638/Paper86.pdf
|volume=Vol-1638
|authors=Denis Shchepakin,Michael Kavanaugh,Leonid Kalachev
}}
==Modeling of ambient glutamate concentration measurement in the mammalian nervous system ==
https://ceur-ws.org/Vol-1638/Paper86.pdf
Mathematical Modelling
MODELING OF AMBIENT GLUTAMATE
CONCENTRATION MEASUREMENT IN
THE MAMMALIAN NERVOUS SYSTEM
D. Shchepakin1 , M. Kavanaugh2 , L. Kalachev1
1
Department of Mathematical Sciences,
University of Montana, Missoula, MT, USA
2
Center for Structural and Functional Neuroscience,
University of Montana, Missoula, MT, USA
Abstract. A neuron is an electrically excitable cell that processes
and transmits information through electrical and chemical signals.
Neurons connect and pass signals to other cells through the struc-
ture called synapse. We focus on synapses through which the signals
are transferred by signaling molecules called neurotransmitters. One
of the predominant excitatory neurotransmitters in the central ner-
vous system of the mammals, including humans, is glutamate. It is
directly or indirectly involved in most brain functions. However, the
excessive stimulation of the glutamate receptors is toxic to neurons,
therefore it is important to rapidly clear the glutamate from the extra-
cellular space and keep its concentration low. Glutamate transporters
play a crucial role in regulating glutamate concentration in synaptic
clefts. Thus, it is important to understand the mechanisms underlying
this process. We describe measurement of the glutamate concentra-
tion in the extracellular space. It is important to estimate the baseline
glutamate concentration to use it in future models and studies. How-
ever, two existing methods of measuring the glutamate concentration
in the extracellular space give inconsistent results with about 100 fold
difference. We construct the model of the process of the glutamate
concentration measurement in order to explain that discrepancy.
Keywords: glutamate transport, ambient neurotransmitter, tonic
signaling, microdialysis
Citation: Shchepakin D, Kavanaugh M, Kalachev L. Modeling of am-
bient glutamate concentration measurement in the mammalian ner-
vous system. CEUR Workshop Proceedings, 2016; 1638: 717-730.
DOI: 10.18287/1613-0073-2016-1638-717-730
Introduction
A neuron is an electrically excitable cell that processes and transmits informa-
tion through electrical and chemical signals. A typical neuron has a cell body,
multiple dendrites and one axon (Figure 1). Because neurons are electrically
excitable, change in a cross-membrane potential could cause the activity of volt-
age dependent ion channels. If this change is high enough, it will cause a one
way electrochemical pulse along the axon, called action potential. When action
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Fig. 1. Neuron and synapse structures
(http://commons.wikimedia.org/wiki/File:Chemical synapse schema cropped.jpg)
potential reaches the end of the axon it will activate the series of reactions, which
transfer the signal to the adjacent cell the axon is connected to. This connection
is called the synapse connection and the connection structure is called a synapse.
The cell that initiates the signal is called a presynaptic cell, and the one that
receives the signal is called a postsynaptic cell. Depending on the type of reac-
tions that process in the synapse to transfer the signal, there are two different
types of synapses: chemical synapses and electrical synapses. We will focus on
the chemical synapses. The end of the axon in the synapse connection contains
vesicles with signaling molecules called neurotransmitters, which are released
into the thin space, called synaptic cleft, between presynaptic and postsynaptic
cells when action potential hits. The postsynaptic cell possesses the receptors
that are triggered by the neurotransmitters, which allows the postsynaptic cell
to that start the cascade of the reactions leading to the further activity.
We will consider the dynamic of the glutamate, which is the predominant excita-
tory neurotransmitter in the central nervous system of the mammals, including
humans. It is directly or indirectly involved in most brain functions. However,
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the excessive stimulation of the glutamate receptors is toxic to neurons, there-
fore it is important to rapidly clear the glutamate from the extracellular space
and keep it low. Glutamate transporters play a crucial role in regulating gluta-
mate concentration in synaptic clefts. Thus, it is important to understand the
mechanism of this process.
Methods of measuring the glutamate concentration in a
brain
The knowledge of the concentration of glutamate in the extracellular space is cru-
cial for applying theoretical models describing its dynamic. Two different meth-
ods exist to estimate this value: microdialysis and electrophysiological methods.
However, microdialysis measurement turns out to be about 100-fold higher than
the electrophysiological measurement [1]–[4]. The reasons for this discrepancy
are yet unknown.
Fig. 2. Schematic illustration of a microdialysis probe
(http://commons.wikimedia.org/wiki/File:
Schematic illustration of a microdialysis probe.png)
Microdialysis is an invasive sampling technique that is widely used to measure
the concentrations of free analyte in the extracellular fluid. The idea is to insert
a microdialysis catheter with semipermeable walls into the tissue (Figure 2). Be-
cause of the perfusion, one is able to measure the concentration of the substance
of interest after some time when the stable state of the system has been reached.
It was suggested that because of the invasiveness of the microdialysis method
the glutamate transporters activity could be reduced in a small region adjacent
to the dialysis probe, which might lead to the overestimation of the glutamate
concentration in the extracellular space.
We will construct a mathematical model of this process under several assump-
tions in order to investigate the dynamic and the steady state of the glutamate
concentration inside the dialysis probe and in a small region nearby. Our goal
is to find a possible explanation for the discrepancy of the two mentioned above
methods using the constructed model.
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Mathematical model
Model assumptions
Before constructing our model we need to formulate all assumptions on which it
will be based.
It is reasonable to assume that glutamate molecules do not appear from or dis-
appear to nowhere. Thus, all changes in glutamate concentration must follow
the conservation law. Moreover, because of the nature of the process, one can
expect the model may be formulated in the form of a diffusion partial differen-
tial equation. Therefore, the first Fick’s law can apply. Furthermore, for the
simplicity of the model we will assume that the diffusion coefficient in this law
is constant.
Now, we discuss some basic assumptions on cells and on glutamate transporters.
To transport any glutamate into the cell, a glutamate molecule must bind to
a transporter. After that the glutamate molecule can either unbind from the
transporter returning the system to the previous state or penetrate inside the cell
via transporter, again unbinding from the transporter. There is an evidence that
binding and unbinding rates of glutamate molecules and glutamate transporters
are much faster than the transfer rate (also called turnover rate) of glutamate
molecules into the cell by the transporters [5]. Additionally, a leak of glutamate
from the cells into the extracellular space by simple diffusion is reported [6]. The
evidence exists that the concentration of the glutamate is much higher inside the
cells than in the extracellular space [7], which suggests that the leak could be
assumed to be constant.
Next, let us put several assumptions about the microdialysis catheter and the
region of the treatment. First, we will assume that the dialysis probe is a perfect
circular cylinder. Moreover, there are no sources or sinks of the glutamate inside
the probe. We assume that only the walls of the catheter are permeable for the
glutamate, that is, there are zero fluxes of glutamate through the top and the
bottom of the cylinder. We will also assume that the conditions inside and
outside the probe are symmetric with respect to the tube axis and along it.
Finally, we will assume that the invasiveness of the treatment caused the re-
duction in transporting rate of the transporters. The closer the cells are to the
location of the insertion, the bigger the reduction in transporting rate. We will
assume, that the cumulative reduction of the transporting rate of all transporters
can be approximated by a Gaussian function as a function of a distance from
the location of treatment.
Model equation derivation
One can imagine that modeling of the dynamics of the glutamate concentra-
tion can be represented as the movement of many individual molecules in some
volume. Let us call this region Ω, the open subset of Rn , where n ≥ 1. Of
course, the most interest for us will be the case n = 3. Moreover, because of the
assumed symmetry of the Ω it is possible to map it on the subset of a space of
lower dimension, but, for now, we will consider the case of n = 3.
Let us introduce more notations, that we will use to construct our mathematical
model.
• Let V be an arbitrary compact volume such that V ⊂ Ω and ∂V be a
piecewise smooth boundary of the region V .
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• The density function u(t, x), where x ∈ Ω is a point in 3-dimensional space
and t ≥ 0 is time. The units of the values of this function correspond to
the volume density, that is, the amount of the glutamate per unit volume.
We will assume, that the density function u(t, x) is sufficiently smooth.
• Let φ(t, x) be the flux of the glutamate at the location x ∈ Ω, at time t ≥ 0.
It measures the amount of the glutamate crossing the point x, at time
t ≥ 0. The units of the values of this function are the amount of glutamate
per unit area, per unit time. One should notice, that φ : R × R3 → R3 ,
that is, the flux function is a vector function.
• Let f (t, x) be the rate at which the glutamate appears or disappears per
unit volume at the location x ∈ Ω, at time t ≥ 0. We will call the function
f a source if it positive, sink if it is negative and source-sink if it takes
both positive and negative values or if the sign of its values is unclear.
Let’s now consider the dynamics of a glutamate amount within the arbitrary
volume V . According to the conservation law, the rate of change of the amount
of the glutamate in that volume must be equal to the rate at which the glutamate
amount flows in through the boundary ∂V minus the rate at which it flows out
through the boundary ∂V plus the rate at which it appears within the volume V
minus the rate at which it disappears within the volume V . Therefore, one can
write this in the mathematical form, using the introduced notations, as follows
Z Z Z
d
u(t, x)dv = − φ(t, x)da + f (t, x)dv. (1)
dt
V ∂V V
Now, applying the Divergence theorem to the first term on the right hand side
of the equation (1), we obtain
Z Z Z
d
u(t, x)dv = − ∇ · φ(t, x)dv + f (t, x)dv.
dt
V V V
Using Leibniz integral rule, we get
Z Z Z
∂
u(t, x)dv = − ∇ · φ(t, x)dv + f (t, x)dv.
∂t
V V V
Next, moving all term to the left hand side and combining all the integrals, we
can write
Z
∂
u(t, x) + ∇ · φ(t, x) − f (t, x) dv = 0. (2)
∂t
V
Because the equation (2) must hold for any volume V , and because the functions
under the integral are continuous, we arrive at
∂
u(t, x) = −∇ · φ(t, x) + f (t, x). (3)
∂t
Fick’s first law, which states that the flux goes from regions of high concentration
to regions of low concentration with a magnitude that is proportional to the
concentration gradient can be written in mathematical form as
φ(t, x) = −D(t, x)∇u(t, x),
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where D is a diffusion coefficient. We will assume, that the diffusion coefficient
doesn’t depend on time and location, that is, D is a constant. Thus, substituting
the flux expression into the equation (3), we obtain
∂
u(t, x) = D4u(t, x) + f (t, x). (4)
∂t
Our next step is to derive the expression for the function f . We can represent
this function as a combination of two processes: one process is related to the
chemical kinetics of the glutamate molecules because of the reacting with gluta-
mate transporters and another is related to the leak of the glutamate molecules
from the cells into the extracellular space. We write
f (t, x) = fck (t, x) + fleak (t, x). (5)
Because of the assumptions we made, that leak is invariable in space and time:
fleak ≡ KL ,
where KL is a constant. Therefore, the equation (5) can be written as
f (t, x) = fck (t, x) + KL . (6)
Let us now consider time dependent chemical kinetics of glutamate molecules and
transporters in an arbitrary volume W ⊂ Ω. To transfer a glutamate molecule
inside the cell, the molecule must bind to the glutamate transporter first to
form a complex. Moreover, we assumed that the bound glutamate molecule
could unbind from and free the transporter before it will transfer the molecule
into a cell. The corresponding chemical kinetics scheme for this process can be
written as follows
k+ kV
Gout + T r −
−
)*
−− C, C −−→ T r + Gin ,
k−
where Gout is extracellular glutamate, Gin is intercellular glutamate, T r is a
transporter and C is a glutamate-transporter complex, k+ and k− are the binding
and unbinding rate constants respectively and kV is the rate of transfer of the
glutamate into a cell (thus, the glutamate goes into the cell and complex becomes
a free transporter).
Applying the Law of Mass Action, we can write the system of the differential
equations that describes those reactions as follows:
d[Gout ]
= −k+ [Gout ][T r] + k− [C],
dt
d[T r]
= −k+ [Gout ][T r] + (k− + V )[C],
dt
(7)
d[C]
= k+ [Gout ][T r] − (k− + V )[C],
dt
d[Gin ]
= V [C],
dt
where [A] is a notation for the concentration of substance A and d[A]
dt is the time
derivative of the concentration of substance A.
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We will denote the initial conditions as
[Gout ](0) = G∗out , [T r](0) = T r∗ , [C](0) = C ∗ , [Gin ](0) = G∗in .
One should notice that the total number of the glutamate transporters and,
therefore, the concentration of the transporters, which is the sum of concen-
tration of free transporters and bound transporters (complexes), should remain
constant. Indeed,
d[T r] d[C]
+ = 0,
dt dt
so
[T r] + [C] = const = [T r](0) + [C](0) = T r∗ + C ∗ = N,
then
[T r] = N − [C]. (8)
Now, according to our assumption, the binding/unbinding rates of the glutamate
molecules to transporters are much faster than the rate at which the glutamate
molecules are transferred inside the cells (via the glutamate transporters). We
can write this assumption in the mathematical form using our notations to get
k+ N ∼ k− kV .
Let’s introduce a small parameter 0 < ε 1 and a new notations
kf
+ = εk+ , k− = εk− .
f
Then we have
+ N ∼ k− ∼ kV .
kf f
Using these new notations, we can rewrite the system (7) in the following form
d[Gout ]
ε = −kf
+ [Gout ][T r] + kf
− [C],
dt
d[T r]
ε = −kf
+ [Gout ][T r] + (kf
− + εV )[C],
dt
(9)
d[C]
ε = kf
+ [Gout ][T r] − (kf
− + εV )[C],
dt
d[Gin ]
= kV [C].
dt
Our next step is to apply Thikhonov’s and Vasil’eva’s theorems [8] to this per-
turbed problem to obtain the reduced version of the original system (7). Because
we are not really interested in the behavior of the system near initial instant of
time, we will consider only the, so called, regular part of the asymptotic ex-
pansion. For the sake of simplicity, we will consider only leading order approx-
imation. Thus, we can represent the solution of the system (9) and, therefore,
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original system (7) as
[Gout ] = [Gout ] + O(ε),
[T r] = [T r] + O(ε),
[C] = [C] + O(ε),
[Gin ] = [Gin ] + O(ε).
Substituting these expressions into the system (9) and considering only equations
of zero order approximation ε, we will have
0 = −kf
+ [Gout ] [T r] + kf
− [C],
0 = −kf
+ [Gout ] [T r] + kf
− [C],
0 = kf
+ [Gout ] [T r] − kf
− [C],
˙
[Gin ] = kV [C],
which leads to
kf
+ [Gout ] [T r] = kf
− [C],
˙
[Gin ] = kV [C].
After substituting (8) into the first equation above, we write
+ [Gout ] (N − [C]) = k− [C],
kf f
and
+ N [Gout ]
kf
[C] = .
− + k+ [Gout ]
kf f
Thus, the reduced system is
N [Gout ]
[C] = ,
Kd + [Gout ]
(10)
˙ [Gout ]
[Gin ] = N kV ,
Kd + [Gout ]
k−
where Kd = .
k+
The second equation of the system (10) describes the leading order approxima-
tion of the glutamate flow rate from the extracellular space into the cells due
to the activity of the glutamate transporters. It depends on the concentration
of transporters N , their turnover rate kV , the binding-unbinding rate constants
k± and the concentration of the glutamate outside the cells. Because of the
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assumptions we made about the resulting transfer rate of all transporters we
can now express function fck (t, x) in the notations of the equation (4) as
u(t, x)
fck (t, x) = J(l) ,
Kd + u(t, x)
where l is a distance between the location x and the axis of the dialysis probe.
Also,
0, 0 ≤ l ≤ L,
J(l) = (d − L)2 (11)
−
2σ 2
J 1 − e , l > L,
max
where Jmax is a cumulative transfer rate coefficient in a healthy tissue, L is a
radius of the microdialysis catheter, and σ is some parameter that corresponds to
the amount of influence of the treatment on the turnover rate of the transporters.
Thus, the equation (4) can be written as
∂ u(t, x)
u(t, x) = D4u(t, x) − J(d) + KL . (12)
∂t Kd + u(t, x)
Because of the assumed symmetry and non-permeability of the top and the bot-
tom of the cylindrical dialysis probe, the equation (12) can be greatly simplified
and reduced to a lower dimension equation.
Fig. 3. The cylindrical probe.
It is more convenient to consider the equation (12) in cylindrical coordinates.
So, we need to change variables using the following expressions, assuming that
x = (x, y, z)
x = r cos φ,
y = r sin φ,
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where r is the Euclidean distance from the z axis to the point x and φ is the
azimuth of the point, that is, the angle between the reference direction on the
xy plane and the line from the origin to the projection of point x on the plane.
Moreover, we make the origin of the coordinate system be in the center of a
lower base of the cylindrical probe with z axis being coinciding to the axis of
the cylinder (see Figure 3). The equation (12) will transform into
∂ u(t, r, φ, z)
u(t, r, φ, z) = D4u(t, r, φ, z) − J(r) + KL , (13)
∂t Kd + u(t, r, φ, z)
where 4u(t, r, φ, z) is a Laplace operator in a cylindrical coordinates acting on
u:
1 ∂2u ∂2u
1 ∂ ∂u
4u(t, r, φ, z) = r + 2 2 + 2.
r ∂r ∂r r ∂φ ∂z
Because of the assumptions we made about symmetry with respect to the axis
of the cylindrical probe and along that axis, the function describing glutamate
concentration will not depend on φ and z, that is,
u(t, r, φ, z) = u(t, r).
Therefore, the Laplace operator in our case will be just
∂ 2 u 1 ∂u
1 ∂ ∂u
4u(t, r) = r = + .
r ∂r ∂r ∂r r ∂r
Now we can rewrite equation (13) as
∂ 2 u 1 ∂u
∂ u
u=D + − J(r) + KL , (14)
∂t ∂r r ∂r Kd + u
where u = u(t, r).
Finally, we need to derive boundary and initial conditions for the constructed
model equation. Again, using the symmetry assumptions, we can state that
there is no flux through the center of the cylindrical catheter. We can write this
as
∂u
= 0. (15)
∂r r=0
Next, let us denote the concentration of glutamate in the extracellular space
by us . We expect the microdialysis treatment to affect the concentration of
glutamate only in some small region, therefore,
u(t, ∞) = us . (16)
The initial conditions for the concentration of the glutamate: some specified
concentration u∗ inside the dialysis tube, which can be zero or not, and us
outside the tube:
∗
u , 0 ≤ r ≤ L,
u(0, r) = (17)
us , r > L.
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Let us write down the final version of constructed model (14), (11), (15), (16),
(17):
2
∂u ∂ u 1 ∂u u
=D + − J(r) + KL ,
∂t ∂r r ∂r Kd + u
0, 0 ≤ r ≤ L,
J(r) = (r − L)2 (18)
−
Jmax 1 − e
2σ 2 , r > L,
∂u
= 0, u(t, ∞) = us ,
∂r r=0
∗
u , 0 ≤ r ≤ L,
u(0, r) =
us , r > L.
This problem cannot be solved analytically because of the nonlinear term in the
right hand side of the diffusion equation. Next, we will consider the numerical
method that will allow us to solve this problem computationally.
Implementation and results
Let us introduce physically realistic constants that will be substituted into the
model equation (18) [9].
L = 1500µm, σ = 200µm,
us = 25nM, Kd = 25µM,
Jmax = 2.5mM sec−1 , KL = 2.5µM sec−1 ,
D = 700µm2 sec−1 .
We will use different initial concentrations of glutamate inside the cylindrical
probe (u∗ ) to find the cylindrical volume around the probe, so that the radius of
the base of the cylindrical volume R is as small as possible, but large enough, so
that it ”can still play” the role of an ”infinity” in our model. We can say that the
value R1 of R is large enough if substituting some R2 > R1 into the model will
not change the values of the solution u(t, r), provided by the numerical method,
on the interval [0, R1 ].
We should notice that we assumed the ”true” concentration of glutamate in the
extracellular space to be 25nM . This values is suggested by the electrophysiology
method of measurement of glutamate concentration.
We ran the simulation for different choices of values of R and u∗ and obtained
the results, shown in Figures 4, 5, and 6.
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Fig. 4. Spatial profiles of glutamate concentration at the initial time t = 0 seconds,
at t = 5400 seconds and at the steady state. The initial concentration in the tube is
u∗ = 10mM . First row corresponds to the choice of R = 3000µm, second row
corresponds to the choice of R = 6000µm.
Fig. 5. Spatial profiles of glutamate concentration at the initial time t = 0 seconds,
at t = 5400 seconds and at the steady state. The initial concentration in the tube is
u∗ = 0M . First row corresponds to the choice of R = 3000µm, second row
corresponds to the choice of R = 6000µm.
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Fig. 6. Function of glutamate concentration at the center of the probe as a function
of time. The initial concentration in the tube was u∗ = 10mM.
We can notice in Figures 4 and 5 that the resulting concentration looks be the
same for R = 3000µm and R = 6000µm. Computations shown no significant
difference between the solutions for those two values of R. Therefore, R =
3000µm can be used to represent ”infinity” in our numerical model. Moreover,
we note that no matter what initial conditions are, the system always reaches
the same steady state. The only difference is the time interval within which
this steady state is reached. Therefore, one could use any initial conditions to
analyze the steady state of the system, given enough time passed. As we can
see on the graphs, the concentration of glutamate at the steady state inside the
microdialysis probe is constant, i.e. the glutamate there is uniformly distributed.
In Figure 6 the time dependence of glutamate concentration at the center of the
cylindrical catheter (r = 0) is shown.
Let us now compare the values of concentration inside and outside of the dialy-
sis catheter at the steady state. The concentration of glutamate outside of the
catheter at ”infinity” is just the glutamate concentration in the extracellular
space us . By our assumption, it’s the concentration provided by the electro-
physiology method, and it equals to 25nM . Under that assumption the model
predicts about 3.5µM for the glutamate concentration inside the catheter, which
is within the range of the usual values estimated via microdialysis method.
To sum up, we constructed a model of the microdialysis method under sev-
eral assumptions. Assuming that the true concentration of the glutamate in
the extracellular space is provided by the electrophysiology method, the model
predicts the values of the microdialysis measurements that match the values ob-
tained by the real measurements. Therefore, we provided possible explanation
for the discrepancy in measurements between microbialysis and electrophysiol-
ogy methods.
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