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 structure 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 nervous 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 extracellular 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 concentration in the extracellular space. It is important to estimate the baseline glutamate concentration to use it in future models and studies. However, two existing methods of measuring the glutamate concentration in the extracellular space give inconsistent results with about 100 fold di erence. We construct the model of the process of the glutamate concentration measurement in order to explain that discrepancy.
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 voltage 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 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 reactions that process in the synapse to transfer the signal, there are two di erent 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.
the excessive stimulation of the glutamate receptors is toxic to neurons, therefore it is important to rapidly clear the glutamate from the extracellular space and keep it low. Glutamate transporters play a crucial role in regulating glutamate concentration in synaptic clefts. Thus, it is important to understand the mechanism of this process.
cial for applying theoretical models describing its dynamic. Two di erent methods exist to estimate this value: microdialysis and electrophysiological methods.
the concentrations of free analyte in the extracellular uid. The idea is to insert a microdialysis catheter with semipermeable walls into the tissue (Figure 2). Because 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.
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 nd a possible explanation for the discrepancy of the two mentioned above methods using the constructed model.
will be based.
It is reasonable to assume that glutamate molecules do not appear from or disappear 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 di usion partial di erential equation. Therefore, the rst Fick's law can apply. Furthermore, for the simplicity of the model we will assume that the di usion coe cient in this law is constant.
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 di usion 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 uxes 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.
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.
tion can be represented as the movement of many individual molecules in some volume. Let us call this region , 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 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.
The density function u(t; x), where x 2 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.
Let (t; x) be the ux of the glutamate at the location x 2 , at time t 0.
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 : R R3 ! R3, that is, the ux function is a vector function.
unit volume at the location x 2 , 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 ows in through the boundary @V minus the rate at which it ows 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 u(t; x)dv = (t; x)da +
f (t; x)dv: Z
u(t; x)dv = (t; x)dv + f (t; x)dv:
Z u(t; x)dv = (t; x)dv + f (t; x)dv: d Z dt
V d Z dt
V u(t; x) + r (t; x) f (t; x) dv = 0:
Z
@
@t u(t; x) = r (t; x) + f (t; x): (
D(t; x)ru(t; x);
where D is a di usion coe cient. We will assume, that the di usion coe cient
doesn't depend on time and location, that is, D is a constant. Thus, substituting
the ux expression into the equation (
KL; f (t; x) = fck(t; x) + KL:
where KL is a constant. Therefore, the equation (
k+ Gout + T r )* C; C k!V 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 di erential
equations that describes those reactions as follows:
where [A] is a notation for the concentration of substance A and dd[At ] is the time
derivative of the concentration of substance A.
(
kf+N k f kV :
[Gout](0) = Gout; [T r](0) = T r ; [C](0) = C ; [Gin](0) = Gin: One should notice that the total number of the glutamate transporters and, therefore, the concentration of the transporters, which is the sum of concentration of free transporters and bound transporters (complexes), should remain constant. Indeed, [T r] + [C] = const = [T r](0) + [C](0) = T r + C = N; kf+[Gout][T r]
(kf + "V )[C];
turbed problem to obtain the reduced version of the original system (
where Kd = k+
tion of the glutamate ow 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 kf+[Gout] (N
[C]) = kf [C]; kf+N [Gout] kf + kf+[Gout]
:
[C] [G_in] = =
N [Gout] Kd + [Gout] ; [Gout] Kd + [Gout] ; N kV k .
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 (
u(t; x) Kd + u(t; x)
; where l is a distance between the location x and the axis of the dialysis probe. Also, 8 > > > > > J (l) = <
0 0; e (d
L)2 1 where Jmax is a cumulative transfer rate coe cient in a healthy tissue, L is a radius of the microdialysis catheter, and is some parameter that corresponds to the amount of in uence of the treatment on the turnover rate of the transporters.
(12)
x = (x; y; z) x =
r cos ; y =
r sin ; where r is the Euclidean distance from the z axis to the point x and 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.
u(t; r; ; z) = D4u(t; r; ; z)
J (r)
u(t; r; ; z) Kd + u(t; r; ; z) where 4u(t; r; ; z) is a Laplace operator in a cylindrical coordinates acting on u: 4u(t; r; ; z) = r 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 and z, that is,
= 0: u(t; 1) = us:
by us. We expect the microdialysis treatment to a ect the concentration of glutamate only in some small region, therefore,
concentration u inside the dialysis tube, which can be zero or not, and us outside the tube: u(t; r; ; z) = u(t; r): Therefore, the Laplace operator in our case will be just 4u(t; r) = r Now we can rewrite equation (13) as
J (r)
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 ux through the center of the cylindrical catheter. We can write this as
8 u ; 0 u(0; r) = < r
L; : us; (13) (14) (15) (16) (17)
Let us write down the nal version of constructed model (14), (11), (15), (16), (17):
J (r)
u = 0; 0; e (r
L)2 1
L us D Jmax = = = = We will use di erent initial concentrations of glutamate inside the cylindrical probe (u ) to nd 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 "in nity" 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].
Fig. 4. Spatial pro les 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 = 3000 m, second row corresponds to the choice of R = 6000 m.
Fig. 5. Spatial pro les 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 = 3000 m, second row corresponds to the choice of R = 6000 m. We can notice in Figures 4 and 5 that the resulting concentration looks be the same for R = 3000 m and R = 6000 m. Computations shown no signi cant di erence between the solutions for those two values of R. Therefore, R = 3000 m can be used to represent "in nity" in our numerical model. Moreover, we note that no matter what initial conditions are, the system always reaches the same steady state. The only di erence 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.
cylindrical catheter (r = 0) is shown.
Let us now compare the values of concentration inside and outside of the dialysis catheter at the steady state. The concentration of glutamate outside of the catheter at "in nity" is just the glutamate concentration in the extracellular space us. By our assumption, it's the concentration provided by the electrophysiology method, and it equals to 25nM . Under that assumption the model predicts about 3:5 M 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 several 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 obtained by the real measurements. Therefore, we provided possible explanation for the discrepancy in measurements between microbialysis and electrophysiology methods.