Analytical Solutions of Nonlinear Differential Equations in the Mathematical Model for Inactivation of Nitric Oxide by Rat Cerebellar Slices

A mathematical model for the inactivation of nitric oxide by rat cerebellar slices under non-steady state condition has been analyzed. This diffusion-inactivation model was used to estimate the ki-netics of NO consumption by the rat cerebellar slices. He's Homotopy perturbation method is used to solve the first order nonlinear differential equations which describe the concentrations given by net of diffusion and inactivation by the slices. Analytical expressions for the concentration of nitric oxide have been derived for all values of parameters. The obtained analytical results are compared with the simulation results (Matlab/Scilab program) and are found to be in good agreement .


Introduction
Nitric oxide is an important biological regulator and is therefore a fundamental component in the fields of neuroscience, physiology and immunology.It is a diffusible free radical synthesized by a family of nitric oxide synthases (NOS) which participates in a wide range of signaling pathways in biological tissues, mediating phy-siological processes such as vasodilation, memory and learning, neuronal development, regulation of immune response among many others [1]- [5].NO acts as an intracellular messenger throughout the brain.In mammals including humans, NO is an important cellular signaling molecule involved in many physiological and pathological processes [6].It has been emerged as a biological signaling molecule, initially in the cardiovascular, nervous and immune systems [7]- [10].
NO can cross cell membranes easily by diffusion and it is a highly reactive and unstable free radical species.Knowledge about the physiological existence of nitric oxide (NO) concentrations is essential for developing a quantitative understanding of NO signaling, for performing experiments with NO that emulate reality and for knowing whether or not NO concentrations become abnormal in disease states.Nitric oxide acts as a novel neural biological messenger and stimulates soluble guanylylcyclase by increasing the levels of cyclic guanosine 3',5'-monophosphate (cGMP) in target cells.This mechanism is believed on by using incubated slices of cerebellum at brain in which the NO/cGMP system has been best characterized.The cerebellum is that part of the brain in which the NO system has been best studied and where the NO synthase is concentrated.It also offers the special experimental advantages of being a simple laminated structure which contains only a few cell types and of having a well-defined synaptic circuitry.
The evidence implicating NO in focal cerebral injury in mice [11] and in 7-day-old rat pups [12] had been reported during last two decades.The production of NO in a variety of intact living systems such as macrophages [13] and cerebellar slices [14]- [16] has been inferred by measuring nitrite (NO 2− ), nitrate (NO 3− ) and cyclic guanosine monophosphate (cGMP) in the absence and also in the presence of potent and selective inhibitors of NOS where nitrite and nitrate are considered as the major stable metabolites of NO under aerobic conditions and NO would activate brain guanylylcyclase, which leads to cGMP production [17] [18].The NO-cGMP pathway has several neurophysiological roles such as synaptic plasticity, neuronal development and the modulation of membrane excitability [19].Excessive production of NO, would cause damage by various mechanisms such as inhibition of metabolic pathways and in the presence of superoxide anions, the formation of toxic free radicals [20] [21].
Incubated rat cerebellar slices were exposed to fixed NO concentrations and the produced cGMP response in the whole slice because of diffusion is used as an indicator of NO concentration.This diffusion-inactivation process is modelled by Hall et al. [22] under different physiological conditions by using ordinary differential equations.In this communication, the differential equations involved in the mathematical model are solved analytically, and the analytical expressions so obtained are used for calculating the quantification of cGMP response by means of predicted curves.This theoretical analysis asserts that by using different experimental values of the parameters involved in this diffusional model, the quantification of inactivation of nitric oxide can be achieved.

Mathematical Formation of the Problem
The overall change in the concentration of NO with respect to time "t" at a particular position in the slice (x) is represented by the net of diffusion and inactivation by the slice.The inactivation process is Michaelis-Menten in nature and it is described by a first order partial differential equation [22].
where [ ] NO is the concentration of nitric oxide, D is the diffusion coefficient of nitric oxide, max V and m K are the Michaelis-Menten parameters.Equation (1) can be written in another way by introducing a pseudo-first order rate constant where The Equation ( 2) is solved for the following initial and boundary conditions By introducing the following dimensionless parameters The governing nonlinear reaction/diffusion Equation ( 2) is expressed in the non-dimensional format as follows: The initial and boundary conditions may be presented as follows: ( ) Analytical solution of Equation (6), by using the initial and boundary conditions in (7) is given by

Stepwise Activation of Neuronal NOS
The equation which is derived from the Michaelis-Menten equation, describes the generation of NO signal at various rates of NO synthesis and is given as follows where 1 v is the rate of NO production and [ ] NO is the concentration of the nitric oxide.The initial condition for solving Equation ( 9) is given by By applying new Homotopy approach in Equation ( 9) and by solving it we get the analytical solution as follows: [ ]( ) ( ) where κ is defined as in Equation ( 3).The decline of NO concentration following cessation of synthesis was derived by the following equation The initial condition for solving the Equation ( 12) is given by The analytical expression obtained by solving the Equation ( 12) using ( 13) is given by where W is the Lambert W function [23].

Dynamic Activation of NOS.
The rate of NO synthesis ( ) v at time t, is given by the product of the maximum activity and two exponential functions as follows: ( ) where 1 k and 2 k are kinetic parameters.Values of maximum synthesis were used to generate NO profiles which can be calculated by finding the net of synthesis and inactivation as given in the following equation.
By solving the above equation analytically using the condition in Equation ( 10), the following solution is obtained: where κ is defined as in Equation (3).

Modelling NO Signals in Three Dimensions
Spatially discrete NO signaling was investigated by modelling diffusion, synthesis and inactivation in three dimensions .The relevant equation which describes the radial diffusion away from a central point, where NO synthesis is positive within boutons and zero outside them is represented as The corresponding initial and boundary conditions are given as follows: where a and b are constants.The Equation ( 17) is reduced into the normalized form by introducing some dimensionless parameters as follows: The dimensionless form of the Equation (17) using Equation ( 19) is given by The boundary and initial conditions in dimensionless form are as follows: 0 when 0 0 when 0 Now by applying Laplace transform technique, method of reduction of order and complex inversion formula for solving (20) using (21), we obtain the following analytical expression for the dimensionless concentration of nitric oxide as

Quantification of cGMP Profile across the Slice
The cGMP concentration at each position across the slice can also be quantified from the Hill equation where and the slope ( ) Calculation of cGMP concentration using the above Equation ( 23) allows the generation of predicted NO concentration-cGMP response curves for number of different values of kinetic parameters for cerebellar slices.From such curves, quantification of inactivation can be achieved by comparing the experimental data based on the reaction diffusion model which is described above.

Numerical Simulation
The non-linear differential Equations ( 6), ( 9), ( 12), ( 15) and ( 20) for the corresponding boundary & initial conditions in ( 7), ( 10), ( 13) and ( 21) are solved by using numerical methods.The function pdex 4 in Matlab software, which is a function for solving boundary value problems, was used to solve these equations numerically and the Matlab programs are given in supplementary material of this manuscript.
Figure 1(a) indicates the concentration of nitric oxide for various values of 1 α and for some fixed value of 1 β and τ .From this time dependent evolution, we infer that the concentration of nitric oxide increases with the decrease in the value of 1 α or the diffusion coefficient decreases.The normalized concentration becomes uniform and the curve becomes straight line when 1 0.025 α ≤ . In this figure the numerical results are compared with the analytical results and it is noted that there is an excellent agreement between the results which are shown by dotted lines and straight lines respectively.β .But we can witness the reverse reaction i.e., if the time increases, the concentration decreases when the values of the parameters 1 α and 1 β are being increased to some extent.V fixed and by increasing the value of m K , it is noted that the concentration of NO also in- creases and it is worth mentioning that after 0.5 minutes there is no significant difference is noted and the concentration attains its steady state value.But in Figure 5(b), the value of m K is fixed and by varying the value of max V , it is understood that along with the decrease in the value of max V , the value of concentration increases   9) and (11).Solid lines represent the numerical solutions whereas the dotted lines represent the analytical solutions.From these figures it is observed that the rapid rise and fall of the NO concentration due to NO synthesis and inactivation implies that the local NO concentration should faithfully follow even very brief periods of NO formation.In Figure 6(a) by increasing the value of max V , it is observed that the produced curves will converge close to the experimental data and slopes will be steepened considerably., & k k v (rate of NO synthesis).Varying the value of m K (0.1 nM -1 nM) by fixing the value of ( ) , a family of curves will be generated which converge around 1 nM as in Figure 6(d).From the Figures 6(a)-(d), it is instructive that the fall of NO synthesis occurs only after 1 minute.
Figure 7(a) represents the radial diffusion of Nitric oxide for various values of the dimensionless parameter γ and for some fixed values of β and t .This observation implies that the shape of the plot changes as the value of the parameter changes.Here it is noted that the concentration increases gradually with the decrease in the value of γ .Hence for small values of γ it can be predicted that the normalized concentration should vary linearly with the reaction diffusion parameter γ and it attains its steady state.In Figure 7(b) the concentration profiles corresponding to a fixed value of γ are presented by keeping time as a constant parameter.Virtually there is a gradual increase in the concentration as the value of the parameter β increases and it becomes 1 for values of 10 β ≥ .In both the Figure 7(a) and Figure 7(b), it is witnessed that there is an excellent agreement between the analytical and numerical results.
In   15) and ( 16).Solid lines represent the numerical solutions whereas the dotted lines represent the analytical solutions.shows that the cGMP response will start to decrease as the value of γ increases.

Conclusion
The analysis which has been done on the inactivation of NO in rat cerebellar slices is theoretical in nature.Time dependent nonlinear differential equations involved in the mathematical model which depict the physiological NO profiles due to stepwise, dynamic inactivation and also the three dimensional radial diffusion of nitric oxide, are solved analytically and a good agreement between theoretical predictions and numerical results is observed for various kinetic parameters.Using these analytical results, the quantification of nitric oxide consumption due to diffusion (parameter α ) in brain cerebellar slices can be done.The theoretical model described here can be used to make number of predictions about NO-cGMP pathways which will be helpful for the future functional studies based on anatomical observations and also should lead to greater understanding of the role of NO in the nervous system.8) and ( 23).

Figure 1 ,
Figures 4-7represent the comparison of analytical results obtained in this work with the numerical results.Upon comparison it is evident that both the results are in good agreement for different values of the reaction and diffusion parameters.

Figure 1 (
b) exhibits the nitric oxide concentration for various values of 1 β and for some fixed value of 1 α and τ .It is understood from the figure that the concen- tration increases along with the increase in the value of 1 β .Here also the comparison of analytical and numeri- cal results shows a good agreement.It is noted that the concentration is equal to 1 for all values of 1 10 β ≥ and it remains same for all values of 1 0.025 β ≤ .Figure 2(a) represents the plot of dimensionless concentration of nitric oxide versus the dimensionless time τ .The figure reveals that the concentration increases as the value of 1β increases for some fixed value of the thickness of the slice x and kinetic parameter 1 α .Change in diffusion occurs only when 1 other than that the concentration remains same.

Figure 2 (
b) shows that the concentration of nitric oxide increases with the decrease in the value of 1 α .From the figure it is evident that the concentration be- comes zero for 1 100 α ≥ and when 1 0.01 α ≤ , it becomes 1.

Figure 3 (
a) and Figure 3(b) stand for the normalized concentration profiles of nitric oxide for various values of the time parameters.It should be noted the concentration increases with the decrease in time for some fixed values of the parameters 1 α and 1

Figure 4 ( 1 v
Figure 4(a) and Figure 4(b) indicate the physiological profiles due to step-wise activation of nitric oxide and describes the generation of NO signals for some fixed value of rate of NO synthesis ( ) 1 v .In Figure 4(a) by keeping the value of m K fixed and varying the value of max V , it is observed that the concentration increases with the decrease in the value of max V but in Figure 4(b) the value of max V is fixed and by varying m K , an increase in the value of concentration is noted as the value of m K increases.Figure 5(a) and Figure 5(b) also present the physiological NO profiles due to step wise activation of nitric oxide and exhibit the decline of NO concentration following cessation of synthesis.In Figure 5(a), by keeping the value of maxV fixed and by increasing the value of m K , it is noted that the concentration of NO also in- creases and it is worth mentioning that after 0.5 minutes there is no significant difference is noted and the concentration attains its steady state value.But in Figure5(b), the value of m K is fixed and by varying the value of max V , it is understood that along with the decrease in the value of max V , the value of concentration increases

Figure 5 (
a) and Figure 5(b) also present the physiological NO profiles due to step wise activation of nitric oxide and exhibit the decline of NO concentration following cessation of synthesis.In Figure 5(a), by keeping the value of max

Figure 4 .
Figure 4. Plot of concentration profiles of Nitric oxide [ ] NO versus time for various values of the parameters using Equations (9) and(11).Solid lines represent the numerical solutions whereas the dotted lines represent the analytical solutions.

Figure 5 .
Figure 5. Plot of concentration of nitric oxide [ ] NO versus time for various values of the parameters using Equations (12) and (14).Solid lines represent the numerical solutions whereas the dotted lines represent analytical solutions when (a) 1 max 0.5, 0.7, 0.9, 1, 1.3 nM & 5 M s m

Figure 6 (
b) and Figure 6(c) exhibit the cGMP response curves produced as a result of dynamic activation of NO for fixed values of m K , max V and for various values of kinetic parameters 1 2 1

Figure 8 (
a) the situation corresponding to 10 γ = and 0.5 ρ = is graphically exhibited.The rapid diffusion of nitric oxide results in steady-state cGMP levels within 1 minute.It is obvious from the figure that the concentration increases steadily with the increase in the value of β .Alternatively in Figure 8(b) presented.Here it is noted that the diffusion of nitric oxide increases with the decrease in the value of γ .

Figure 6 .
Figure 6.(a)-(d) Plot of concentration profiles of Nitric oxide [ ] NO versus time for various values of the kinetic reaction and diffusion parameters using Equations (15) and (16).Solid lines represent the numerical solutions whereas the dotted lines represent the analytical solutions.

Figure 9 (
Figure 9(a) and Figure 9(b) represent the family of NO concentration-cGMP response curves that have been generated by using Hill's equation and by varying the values of the reaction and diffusion parameters β and γ .From Figure 9(a) it is evident that the cGMP concentration decreases with the value of β and Figure 9(b)shows that the cGMP response will start to decrease as the value of γ increases.

Figure 9 .
Figure 9. Plot of dimensionless cGMP concentration versus time for various values of the parameters & β γ