- Open Access
Computational simulation of a new system modelling ions electromigration through biological membranes
© Alaa and Lefraich; licensee BioMed Central Ltd. 2013
- Received: 15 May 2013
- Accepted: 27 August 2013
- Published: 5 September 2013
The interest in cell membrane has grown drastically for their important role as controllers of biological functions in health and illness. In fact most important physiological processes are intimately related to the transport ability of the membrane, such as cell adhesion, cell signaling and immune defense. Furthermore, ion migration is connected with life-threatening pathologies such as metastases and atherosclerosis. Consequently, a large amount of research is consecrated to this topic. To better understand cell membranes, more accurate models of ionic flux are required and also their computational simulations.
This paper is presenting the numerical simulation of a more general system modelling ion migration through biological membranes. The model includes both the effects of biochemical reaction between ions and fixed charges. The model is a nonlinear coupled system. In the first we describe the mathematical model. To realize the numerical simulation of our model, we proceed by a finite element discretisation and then by choosing an appropriate resolution algorithm to the nonlinearities.
We give numerical simulations obtained for different popular models of enzymatic reaction which were compared to those obtained in literature on systems of ordinary differential equations. The results obtained show a complete agreement between the two modellings. Furthermore, various numerical experiments are presented to confirm the accuracy, efficiency and stability of the proposed method. In particular, we show that the scheme is unconditionally stable and second-order accurate in space.
- Reaction-diffusion system
- Nonlinear coupled system
- Finite element method
- Nernst-Planck equations
- Numerical analysis
- Enzyme kinetics
- Substrate suicide
- Cooperative phenomena
- Computational simulation
Cell membrane is the biological membrane separating the intracellular environment from the extracellular one. The cell membrane surrounds all cells and it’s selectively permeable, permitting the free passage of some substances and restricting the passage of others, thus controlling the flux of substances in and out of the cell. All diseases are problems of regulating the passage of materials at the level of the cell. Consequently, to understand the cause of a disease, we need to understand the alterations that take place at the cellular level. Thanks to mathematical modeling, cell biology phenomena may be expressed by ordinary differential equations or systems of partial differential equations.
An important class of models in cell biology, is the class modeling ion transport through biological membranes. This transport phenomena occurs in most living cells and some biochemical processes. The first models in the literature included one ordinary differential equation for each ion concentration. All this models were based on the implicit assumptions that chemical concentrations are uniform in space. This assumption is reasonable when the region of space where the reaction occurs is confined and very small. Also this models assumed that the electric field is constant inside the membrane.
Despite that the constant electrical field assumption has the advantage of leading to a simple mathematical analysis, all cells maintain a difference in electrical charge across their membrane. This difference in charge give rise to a voltage difference, or electrical potential. Furthermore, there are numerous situations in which chemical concentrations are nonuniform in space. In this sense, we need to establish and compute more accurate mathematical models of ions electromigration through biological membranes.
In this paper, we present the numerical simulation of a more realistic model of ions electromigration through biological membranes. This model is more general than those in literature of membrane transport as it extends them in four topics: 1) it’s a multidimensional model, 2) it doesn’t rely on the constant electrical field assumption, 3) it consider both the temporal and spatial dependence, 4) it includes different reaction kinetics terms.
where Q T =]0,T[×Ω, , T>0; Ω is an open regular set of which represent the biological cell and ∂ Ω represent the cell membrane.
Further information about the modelling of the problem and its mathematical analysis, can be found in .
In the biophysical literature, the early works on these models were interested in the stationary case of passive migration (i.e., without reaction); see [2, 3]. In all these works two popular simplifications were considered, namely the Goldman hypothesis where the electrical field is supposed to be constant inside the membrane and the electroneutral hypothesis where the neutrality at each point of the membrane is assumed; see for example . Mcgillivray  recognized that these models are the limit of the full equations when the ratio of the Debye length to the membrane thickness goes to, respectively, infinity or zero. Usually enzymes are held to biological membranes and ions undergo biochemical reactions when crossing the membrane. Valleton  did a general biophysical study of coupling of electromigration diffusion with biochemical reactions.
In this paper we present a numerical simulation of such systems, for a large class of reaction kinetics, including the usual biochemical kinetics as the Michaelis-Menton one (a mathematical analysis of the one dimensional and stationary case was done by  then we did the mathematical analysis of the multidimensional unsteady case ). This article is organized in the following way. The next section is devoted to finite element discretisation of the mathematical model. Then, we present applications, results and numerical experiments showing the accuracy, efficiency and stability of the proposed method. Finally, conclusions are drawn in the last section.
We represent the solutions as for i s =1,…,N s and with α j (t)=0 for j∈Y0. Then we set for i s =1,…,N s and ζ ϕ =ζ ϕ (t)=(α j (t))1≤j≤m.
We have a nonlinear term due to , we have dealt with according to the model and thus to the expression of .
In this section we present three numerical applications of ions electro-migration through biological membranes. The models of the basic enzyme reaction, the suicide substrate reaction and the cooperative reaction, are numerically simulated.
Result 1 : Enzyme kinetics (basic enzyme reaction)
Algorithm of resolution
● Loop over n
At step n:
● Calculate solution of:
where eps is the stopping criterion.
Here we present changes in substrate, product and enzyme concentrations. The cell is represented by an ellipse with semi-major axis a=2 and semi-minor axis b=1. The diffusion coefficient of the ions are d1=10−3m2.s−1, d2=2.10−3m2.s−1 and d3=5.10−3m2.s−1. The constants of reaction are . The charge number of the ions are z1=1, z2=0 and z3=1. The electric charge density is f = 0.1C. The initial conditions are C1,0=1μ M, C2,0=800 μ M, C3,0=0 and ϕ0=−80 m V; the stopping criterion is e p s = 10−4. The time step of the simulation is d t = 10−3s.
Two simplifications of these equations have been quite popular in the literature while computing membrane reactions, firstly the Goldman hypothesis where the electrical field is supposed to be constant inside the membrane and secondly considering a system of ordinary equations depending only on time and not the space. The added value of this work is not considering all of those simplifications which leads to a more realistic model and more accurate numerical results. Moreover, the results obtained are in agreement with the experimental results found in the literature .
Result 2 : Suicide substrate kinetics
where E, S and P stand for enzyme, substrate, and product, respectively; X and Y, enzyme-substrate intermediates; E i , inactivated enzyme; and the k ’s are positive rate constants.
Algorithm of resolution
● Loop over n
At step n:
Here we present changes in substrate, product, enzyme, inactivated enzyme and the intermediate concentrations (X and Y). The cell is represented by an ellipse with semi-major axis a=2 and semi-minor axis b=1. The diffusion coefficients of the ions are d1=10−3m2.s−1, d2=2.10−3m2.s−1, d3=5.10−3m2.s−1, d4=10−3m2. s−1, d5=2.10−3m2.s−1, d6=4.10−6m2.s−1, the reaction parameters are k1=2 s−1, k−1=4 s−1, k2=12 s−1, k3=10 s−1 and k4=2 s−1. The charge number of the ions are z1=1, z2=0, z3=1, z4=1, z5=1 and z6=0. The electric charge density is f = 0.1C. The initial concentrations are e0=0.5μ M and s0=0.5μ M;and ϕ0=−80 m V. The time step of the simulation is d t = 10−2s. The data employed for the reaction parameters and initial concentrations were taken from Burke et al. .
To highlight the accuracy of these results, we compared them first with the numerical solutions and the approximate asymptotic solutions obtained both by Burke et al. , they considered a system of ordinary differential equations depending on time as they neglected the spatial aspect of the biochemical reaction, and they supposed that the electrical potential inside the membrane remains constant. For the numerical solutions Burke et al.  solved the system numerically, but for the approximate asymptotic solutions they non-dimensionalise the same system, and used asymptotic methods and a method detailed in Kevorkian and Cole . Finally we compared our results with previous approximate methods of Tatsunami et al.  and Waley  which were based on a pseudo-steady state hypothesis. This comparison shows that the results described here are valid numerical solutions for the kinetics of suicide substrate system. The solution for the substrate and inactivated enzyme are more accurate than those of previous approximations [14, 17], especially in small time, which is by definition ignored by any pseudo-steady-state approximate method. Furthermore, the method presented here is specially useful in estimating the intermediate (X and Y) concentrations besides incorporating the spatial and the electro-migration aspects of the phenomena.
Result 3 : Cooperative phenomena
An enzyme reaction is said to be cooperative if a single enzyme molecule, after binding a substrate molecule at one site can then bind another substrate molecule at another site. Such phenomena are quite common in living organisms. Another interesting cooperative reaction is when an enzyme with several binding sites is such that the binding of one substrate molecule at one site can affect the activity of binding other substrate molecules at another site. This indirect interaction between distinct and specific binding sites is called allostery, and an enzyme displaying it, an allosteric enzyme. When a substrate that binds at one site increases the binding activity at another site then the substrate is called an activator, otherwise (if it decreases the activity) it’s called an inhibitor.
Algorithm of resolution
● Loop over n
● At step n:
● Calculate solutions of:initialize for i = 1,…,5,
An enzyme can also exhibit negative cooperativity, in which the binding of the first substrate molecule decreases the rate of subsequent binding. This can be modeled by decreasing k3. We used for positive cooperativity K1=1000, K2=0.001 and K1=0.5, K2=100 for negative cooperativity (this values were taken from ).
The convergence of the basic enzyme reaction
Convergence results for the basic enzyme reaction
h1 = 0.3
h2 = 0.1
h3 = 0.05
The convergence of the suicide substrate reaction
Convergence results for the suicide substrate reaction
h1 = 0.3
h2 = 0.1
h3 = 0.05
The convergence of the cooperative reaction
Convergence results for the cooperative reaction
h1 = 0.3
h2 = 0.1
Now, let us give some information about the numerical stability of our algorithms. We perform a numerical experiment with different time step dt, . These results suggest that the scheme is indeed unconditionally stable as the solutions are quasi the same for different time steps. To illustrate, we chose to represent the product concentration.
Stability of the basic enzyme reaction
Stability of the suicide substrate reaction
Stability of the cooperative reaction
In this paper, a new model simulating ions electro-migration through biological membranes is proposed by using a more general mathematical model and a numerical technique based on the finite element method. The results presented here demonstrate that the model’s behavior agrees with the behavior of biochemical reactions as it’s consistent with the physical interpretation of the phenomena. Moreover, after comparison we can observe a complete consistency with literature findings [9, 13–15, 18]. A variety of numerical experiments were presented to confirm the accuracy, efficiency, and stability of the proposed method. In particular, the scheme was shown to be unconditionally stable and second-order accurate in space.
We are grateful to the anonymous referee for the corrections and useful suggestions that have improved this article.
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