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Mathematical Modelling and Numerical Solution of Chemical Reactions and Diffusion of Carcinogenic Compounds in Cells

DONALD O BESONG

Master’s Degree Project Stockholm, Sweden 2004

TRITA-NA-E04152

Numerisk analys och datalogi KTH 100 44 Stockholm

Department of Numerical Analysis and Computer Science Royal Institute of Technology SE-100 44 Stockholm, Sweden

Mathematical Modelling and Numerical Solution of Chemical Reactions and Diffusion of Carcinogenic Compounds in Cells

DONALD O BESONG

TRITA-NA-E04152

Master’s Thesis in Numerical Analysis (20 credits) at the Scientific Computing International Master Program, Royal Institute of Technology year 2004 Supervisor at Nada was Michael Hanke Examiner was Axel Ruhe

Abstract In order to shed more light on how cancer is triggered, Professor Bengt Jernstrom and his research group at Karolinska Institute (KI) have been performing in vitro incubation of carcinogenic compounds with cells. In vitro reactions and diffusion take place when the carcinogenic substrate is added to cells in culture. Only one cell and its appropriate quota of the medium is needed for the mathematical model, and indeed only a 22.5o sector of a cell is modelled. FEMLAB is the software used for the simulation. The graphical representation of the problem and its simulation is made possible by applying the mathematical technique of homogenisation in the multi-compartment cytoplasm. All constants and parameters used in the simulation were the same used for the in vitro experiments. The model, and consequently the programme, can be adapted to various physical and chemical scenarios. The concentration of the carcinogenic substrate in the extracellular solution is computed, and its half-life is compared to the in vitro results. Both results are found to be the same. The model can be used for the prediction of the experimental inaccessible concentration profile in the nucleus.

Matematisk modellering och numerisk lo¨ sning av reaktioner och diffusion f¨or cancerogena a¨ mnen i celler

Sammanfattning F¨or att belysa hur cancer uppkommer, har prof Bengt Jernstr¨om och hans forskargrupp p˚a Karolinska Institutet (KI) utf¨ort in vitro odling av cancerogena a¨ mnen i celler, d¨ar reaktioner och diffusion d˚a a¨ ger rum. Endast en cell beh¨ovs f¨or att s¨atta upp en matematisk modell, och av denna cell modelleras endast en 22.5 graders sektor. FEMLAB har anv¨ants f¨or simuleringen. Den grafiska representationen av problemsimuleringen har m¨ojliggjorts genom att applicera homogenisering

p˚a multi-compartment cytoplasma. Alla konstanter och parametrar som anv¨ants i modellen hade samma v¨arden som i in vitro experimenten. Modellen, och a¨ ven programmet, kan anpassas till olika fysikaliska och kemiska scenarier. Koncentrationen av de cancerogena a¨ mnena i modellen och deras halvtids livsl¨angder ber¨aknas och j¨amf¨ors med in vitro resultat. B˚ada resultaten o¨ verensst¨ammer. Modellen kan anv¨andas f¨or prediktion av om¨atbara koncentrationer i cellk¨arnan.

Contents 1

Introduction

1

2

The Physical Problem and its Mathematical Model 2.1 Diffusion . . . . . . . . . . . . . . . . . . . . 2.2 Reaction . . . . . . . . . . . . . . . . . . . . . 2.3 Initial Conditions . . . . . . . . . . . . . . . . 2.4 Boundary Conditions . . . . . . . . . . . . . .

. . . .

3 4 4 5 5

3

Scaling and Reformulation 3.1 The diffusion reaction model equation . . . . . . . . . . . . . . .

7 8

4

Simplification of problem by means of homogenisation 4.1 Finding Deff . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Weighted arithmetic mean of the diffusion coefficient . . . 4.1.2 Weighted harmonic mean diffusion coefficient . . . . . . 4.1.3 Decision on Deff for the cytoplasm . . . . . . . . . . . . 4.2 Partition coefficient between homogenised cytoplasm and other subdomains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Solving for concentration; Fraction of C undergoing chemical reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Reaction. Fraction of concentration affected by chemical reaction in the cytoplasm. . . . . . . . . . . . . . . . . .

9 10 10 11 12

5

Model Implementation with FEMLAB 5.1 The Femlab Software . . . . . . . . . . . . . . 5.2 The Geometry . . . . . . . . . . . . . . . . . . 5.3 Subdomain properties, equations and constants 5.4 Constants and Parameters . . . . . . . . . . . . 5.5 Subdomains and their properties . . . . . . . . 5.6 Initial condition . . . . . . . . . . . . . . . . . 5.7 Boundary conditions . . . . . . . . . . . . . . 5.8 Solving . . . . . . . . . . . . . . . . . . . . .

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5.9 6

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Conclusion

20 22

References

24

List of Abbreviations

25

Chapter 1 Introduction The aim of this work is to develop a mathematical model for the in vitro chemical reactions and diffusion of carcinogenic compounds in cells, and then create a computer programme based on the model. Consequently, by using different parameters the programme will enable us to carry out these experiments virtually, and thus predict the risk of cancer in living humans and animals. The primary carcinogenic substrate, in the form of diol epoxides (C), is initially outside the cell. Outside the cell, some of this substrate reacts with water to form tetrols (U ), which do not cause cancer, although it diffuses everywhere in the cell. When the remaining substrate C diffuses through the outer membrane of the cell, it still reacts with water within the cytoplasm, while some of it is converted into glutathione conjugates (B) by an enzyme called GST . B does not cause cancer either, but remains in the cytoplasm, where it is pumped out. The remaining C will reach the nucleus, where it still reacts with water in the nucleus to form U , as well as with DNA to form DNA adducts (A). It is this A that causes cancer. It will stay in the nucleus without diffusing into the other parts of the cell. Each cell in the cell culture used in the study is surrounded by about 168 times its volume of medium. The substrate was then added in the medium for the reactions and diffusion to begin. There are no reactions in the membranes, or so-called lipid compartment of the cell. Reaction only takes place in the aqueous compartment of the cell: In the cytoplasm the reaction of substrate C with water is slow but the enzymatic reaction is much faster. In the nucleus, the reaction of C with DNA is the same rate as the reaction with water in the cytoplasm. Figure 1.1 illustrates the diffusion and reactions in and out of a cell. The partition coefficient, K p , which is the equilibrium ratio of the concentration C or U between any aqueous compartment of the cell and its neighbouring lipid compartment, is 1 × 10−3 . K p depends on the substrate. The complexity of this work lies on the complex geometry, as well as the fact

1

Figure 1.1. Illustration of reactions and diffusion in a cell

that there are many reactions with different products, and their diffusion, in each part of the cell. With mass transfer between various parts of the cell, the work has proved to be more complex, but the bottle-neck is a geometry with heavily varying details. This is handled by applying scaling and homogenisation. We shall go into this after studying the physical domain as presented in chapter 2, which reveals the cytoplasm as consisting of thousands of interconnected tiny parts of the lipid compartment sandwiched between very tiny parts of the aqueous compartment. Chapter 2 sets the physical problem and formulates its mathematical model. It is most important to work with a dimensionless model, so in Chapter 3, the model is scaled and reformulated. In Chapter 4, the homogenisation of the cytoplasm is explained, as well as how the solution of the modified problem is obtained with a knowledge of equilibrium isotherms [7, pp.407, 408]. A brief presentation of the FEMLAB software is found in Chapter 5, where the implementation of the model is described. Also in Chapter 5, there is a list of the variable names in the programme, with their physical meanings, so that users of the programme may easily adapt it to their own needs. The in vitro experiments were performed by Kristian Dreij, who is presently a PhD student of toxicology under the supervisors prof. Bengt Jernstrom and prof. Ralf Morgenstern at K.I. They all, as well as my supervisor at KT H doc. Michael Hanke, have always been quick to help me in this work.

2

Chapter 2 The Physical Problem and its Mathematical Model The concentration of the cells in the culture is one cell per 168 times its volume of the medium. Therefore, the smallest useful part of the physical domain for this study is a cell surrounded by about 168 times its volume of the substrate solution. Originally, the substrate C is found only in this extra-cellular solution. The cell consists of the cell membrane which has a thickness of 0.16 percent of the radius of the nucleus. Beyond the cell membrane is the cytoplasm which has a thickness of 3 times the radius of the nucleus, and the nuclear membrane which has the same thickness as the cell membrane. At the centre of the cell we have the nucleus, 4.8 × 10−6 m in radius. We shall call the outer and nuclear membranes, together with the inner membranes within the cytoplasm, the lipid compartment of the cell, and the rest of the cell the aqueous compartment, which comprises both the cytoplasm and nucleus minus any membranes therein. Note that this aqueous compartment of the cell is not the same as the extracellular water. The cytoplasm contains thousands of inter-connected membrane-like sheets, as well as tiny structures consisting of membranes of the same thickness as the outer membrane. Examples of these structures are the endoplasmic reticulii and the mitochondria. Some of these structures are closed and oval in shape, but contain a aqueous compartment. The nucleus is free from any membranes. However, the ratio of all the lipid compartment of the cell to the volume of the whole cell is about 25 : 100. That is, one quarter of the cell is made up of these membranes. Moreover, since the scaled radius of the whole cell is 4, the scaled thickness of the cytoplasm 3, and the radius of the nucleus 1, and all these membranes are found in the cytoplasm except the cell membrane and the nuclear membrane which can be neglected, the volume fraction of the lipid compartment in the cytoplasm is given

3

Figure 2.1. Schematic picture of the cell showing cytoplasm and nucleus

by 43 0.25 3 4 − 13 



= 0.254

(2.1)

The cytoplasm is therefore packed with membranes, as can be seen in Figure 2.1. This is a diffusion/reaction problem. There is no advection. I will first describe the diffusion modelling and then the reaction modelling. Finally, I will combine them.

2.1

Diffusion

Let us call the ith subdomain Ωi , and let us assume that the concentration of the diffusing species in Ωi is φ. Let the diffusion coefficient of φ in Ωi be D. Then, the rate of change of φ in Ωi is given by ∂φ = D 52 φ ∂t

2.2

(2.2)

Reaction

In this study, C is the only compound which reacts. Let the concentration of C, B, U and A in Ωi be denoted by Ci , BiUi and AiCi respectively. There are three reactions: 4

Ci + water −→ Ui at rate kiCi

(2.3)

where ki is the rate constant of the reaction with water in the ith subdomain subdomain. Ci + GST −→ Bi at rate kBiCi · GST

(2.4)

where the enzyme GST has so many active sites that the reaction rate is always constant. Ci + DNA −→ Ai at rate

kAiCi

(2.5)

kAi and kBi are the rate constants of the reaction with A and B respectively in the ith subdomain subdomain. If a certain reaction does not take place in some subdomain Ωi , then the appropriate reaction rate is simply 0. For example, there are no reactions in the lipid compartment of the cell (membranes), so all the reaction rates there are 0. For instance, in the nucleus Ωnu , the models for the reactions represented by (2.3) and (2.5) are: dCnu = − (knu + kAnu )Cnu (2.6) dt where the subscript nu indicates that the variable belongs to the nucleus.

2.3

dUnu = +knuCnu dt

(2.7)

dAnu = +kAnuCnu dt

(2.8)

Initial Conditions

In all the subdomains, the initial concentration is 0 for all the substances C,U,B, and A, except that the initial concentration of C in the rectangular domain Ω 0 representing the extracellular solution, is C0 .

2.4

Boundary Conditions

On the left boundary of the rectangular subdomain we have a homogeneous boundary condition because of symmetry. On its upper and lower boundaries we also have homogeneous boundary condition because there are no concentration gradients along their normals. The upper and lower boundaries of the sector-like geometry to the right of the rectangular subdomain we also have zero Neumann boundary 5

conditions because there are no concentration gradients in the direction of their normals. Hence, for any species φi at those boundaries, the flux is given by: n · (D 5 φi ) = 0

(2.9)

where n is the normal to the boundary. So far, I have mentioned only external boundaries. I shall consider internal boundaries now: The first internal boundary from the left is that between the rectangular subdomain and the curved boundary. In reality, the whole domain is continuous here, but I decided to split them because of the different scaling factors applied in the rectangular subdomain in the radial direction. To avoid polar coordinates, and taking advantage of the fact that there are no concentration gradients in the θ-direction, I have straightened that subdomain into a rectangle, as opposed to its original arc shape. Then I introduced coupling variables between its right boundary and the left boundary of the remaining part of the geometry. Generalising, if a certain species φi only stays within a certain subdomain Ωi and does not diffuse through, then (2.9) holds for that species. However, if the species diffuses through the boundary separating say Ω1 and Ω2 , then the flux will be a function of φ1 and φ2 at the boundary. Moreover, if the partition coefficient for φ is K p between Ω1 and Ω2 , where K p < 1, then the flux into Ω1 is given by n · D 5 φi = M (φ2 − K p φ1 )

(2.10)

and that into Ω2 through that boundary is simply the negative of the flux into Ω1 . M is the mass transfer coefficient, and it is a measure of the resistance to the transport of any given species between the two given subdomains. A high M implies a small resistance to mass transfer, and vice-versa.

6

Chapter 3 Scaling and Reformulation The scaled domain is seen in Figure 3.1 below.

Figure 3.1. The computational Domain

The scaling was done as follows: x˜ =

x S1

(3.1)

y˜ =

y S1

(3.2)

7

where (x, ˜ y) ˜ are the new coordinates of the computational domain, and S is the scaling factor. For all sub-domains within the cell itself, the scaling factor S 1 = 2.24−6 is the radius of the nucleus. For the rectangular domain, scaling by only S 1 makes its radius to be 18, which is more than four times the radius of the whole cell. To make it graphically convenient, it is again scaled by yet another scaling factor, in order to reduce its thickness to 0.5. Therefore, for the rectangular sub-domain in Figure 3.1, the scaling factor is S2 = 36 × S1 . The concentration of any given species are also scaled: φi φ˜i = C0

(3.3)

where C0 is the initial concentration of C in the water surrounding the cell.

3.1

The diffusion reaction model equation

Applying the above scaling, and combining diffusion and reaction, the general equation for any given subdomain is given by ∂φ˜i D 2 ˜ = 5 φi + Fφ˜i ∂t S

(3.4)

where φ˜ is any of the scaled species, S = S1 or S2 the scaling factor for space, and F is the reaction term representing the rate of change of the scaled concentration due to reaction in that subdomain. Let us again take the example of the nucleus. Then φi is given by D ∂φ˜nu (3.5) = 2 52 φ˜ nu + Fφ˜nu ∂t S1 where φ is any of the species’ concentration present in the nucleus, namely C, U or A, and F is the right-hand side of equations (2.6), (2.7), or (2.8), depending on the appropriate species, and the tilde sign simply means it is normalised by the scaling factor C0 .

8

Chapter 4 Simplification of problem by means of homogenisation The computational domain in Figure 5.1 represents a cell whose cytoplasm is homogeneous, rather than one which has many tiny membranes. Therefore, making the cytoplasm homogeneous would be a wise idea. This method is know as homogenisation. In reality, the cytoplasm is extremely densely packed with lipophilic compartments, in the form of endoplasmic reticulii, mitochondria, etc. Such a set-up is similar to a porous medium [7, p.5]. Therefore, the cytoplasm is a multicompartment medium Ω consisting of two componets: the lipophilic or fatty domain Ω f and the hydrophilic or watery domain Ωw . We then assume that the mixture is homogeneous, i.e. a representative elementary volume (REV ) taken anywhere in Ω is identical. Let ρf =

total volume of Ω f total volume of Ω

(4.1)

and

total volume of Ωw (4.2) total volume of Ω Then the following steps are taken in order to model what happens in the cytoplasm:
• finding an effective diffusion coefficient Deff for the homogenised cytoplasm ρw = 1 − ρ f =

• finding a new partition coefficient between the other parts of the cell and the homogenised cytoplasm • using the above to get the concentration of any diffusing species φ for every point in the homogenised cytoplasm. 9

• applying an appropriate isotherm [7, pp.407, 408] to get the part of the concentration involved in reaction, since only the C within the aqueous compartment reacts, remembering that no reaction takes place in the lipid compartments or membranes.

4.1

Finding Deff

The diffusion path in this model is radial, directed from the cell membrane to the nucleus. If all the membranes in the cytoplasm were oriented parallel to the diffusion path, then Deff = Dar , where Dar is the weighted arithmetic mean diffusion coefficient. If all the membranes in the cytoplasm were oriented perpendicular to the diffusion path, then Deff = Dha , where Dha is the weighted harmonic mean diffusion coefficient. In principle, in any intermediate case, Dar ≤ Deff ≤ Dha or Dar ≥ Deff ≥ Dha [6, p.10].

4.1.1

Weighted arithmetic mean of the diffusion coefficient

The weighted arithmetic mean is obtained if all the Ω f subdomains are oriented parallel to the diffusion path. That is, they are radially oriented in the cytoplasm. If we take a REV containing one Ω f and one Ωw subdomain, then the rate of change of the concentration of any species φ is given by ∂φ ∂φ(w) ∂φ( f ) = + (4.3) ∂t ∂t ∂t where the subscripts w and f denote the watery and fatty subdomains, respectively. If the total volume of this REV is V , we can derive an average concentration for it as follows: Z Z Z 1 1 1 φdv = φ(w) dv + φ dv (4.4) φ= v v v v v v (f) But since φ(w) is non-zero only in Ωw , and φ( f ) non-zero only in Ω f , (4.4) becomes ∂φ 1 = ∂t v

Z

1 ∂φ dv = v v ∂t

Z

vw

Z ∂φ( f ) ∂φ(w) 1 dvw + dv f ∂t v v f ∂t

(4.5)

where vw and v f are the volumes of the Ωw and Ω f subdomains respectively. But the rate of change of concentration is given by (2.2). Therefore if the average quantity 5φ is 5φ, and hence the average quantity 52 φ is 52 φ, then (4.5) becomes ∂φ 1 = ∂t v

Z

v

Deff

52 φdv

1 = v

Z

vw

Dw

10

52 φdv

1 w+ v

Z

vf

D f 52 φdv f

(4.6)

and since 52 φ is constant over the REV just mentioned,   ∂φ 1 1 1 1 1 2 2 2 2 = Deff 5 φ · v = Dw 5 φ · vw + D f 5 φ · v f = 5 φ · Dw · vw + D f · v f ∂t v v v v v (4.7) Applying (4.1) and (4.2), equation (4.7) becomes

and finally

4.1.2


∂φ = Deff 52 φ = 52 φ · ρw Dw + ρ f D f ∂t

(4.8)

Deff = ρw Dw + ρ f D f = Dar

(4.9)

Weighted harmonic mean diffusion coefficient

The weighted harmonic mean is obtained if all the Ω f subdomains are oriented normal to the diffusion path. That is, they are in series with the Ωw subdomains. If we take a REV containing one Ω f and one Ωw subdomain, then the rate of change of the average concentration of any species φ is given by ∂φ = D · 52 φ ∂t

(4.10)

In this case in series, 52 φ is considered separate for Ωw and Ω f . i.e 52 φ = 5 2 φ w + 5 2 φ f Also,

(4.11)

52 φw = D−1 w ·

∂φ ∂t

(4.12)

52 φ f = D−1 f ·

∂φ ∂t

(4.13)

and

where ∂φ ∂t is for both Ωw and Ω f . Combining (4.10), (4.11), (4.12) and (4.13), Z Z Z  1  2 ∂φ 1 ∂φ 1 −1 2 D−1 dv f 5 φw + 5 φ f dv = Dw dvw + v v ∂t v vw ∂t v v f f or  ∂φ  −1 −1 2 5 φ= ρw · D w + ρ f · D f ∂t Therefore −1 −1 D−1 eff = ρw · Dw + ρ f · D f or Deff =



ρw · D−1 w +ρf 11

· D−1 f

−1

= Dha

(4.14)

(4.15) (4.16) (4.17)

4.1.3

Decision on Deff for the cytoplasm

The thin sheet-like membranes in the cytoplasm may take any orientation. Let us now consider the 3D cell to have its membranes in any one of three orientations: perpendicular to the diffusion path, radially oriented but vertical, or radially oriented but horizontal. This is a perfect, un-biased orientation of the membranes, with one-third of the membranes in each direction. Therefore one-third of the membranes are in series with the aqueous compartment of the cytoplasm, while two-thirds is in parallel, with respect to the diffusion path. Therefore

Deff =

a · Dha + b · Dar a+b

(4.18)

where a = 1 and b = 2. Therefore, depending on what fraction of the membranes we think are oriented in each direction vis-a-vis the three above-mentioned directions, a and b can be changed. However, I and the professors have thought that the unbiased mode of orientation is most natural. Magnified 3D electron micrographs of the cell depict such an orientation of the membranes. Note that the diffusion coefficient along a membrane is not the same as across it. Therefore, in calculating Dha , a different D is used than when calculating Dar . This is implemented in the application.

4.2

Partition coefficient between homogenised cytoplasm and other subdomains

Now that the cytoplasm has been homogenised, all its physical properties have changed. We know the partition coefficient for the species C and U , between Ω w and Ω f is K p < 1. This implies that at equilibrium, Cw = K p ·C f and Uw = K p ·U f . In other words, the concentration of either of those species in Ω f is K p−1 times greater than in Ωw . We now have to remember that the neighbouring subdomains to the homogeneous cytoplasm are in Ω f : viz the outer membrane and the nuclear membrane. If we denote the partition coefficient between this homogenised cytoplasm and any Ω f by Kˆp , then Kˆp can be derived. With little arithmetics, we have

Kˆp =

1 · ρw + K p−1 · ρ f K p−1 12

(4.19)

4.3

Solving for concentration; Fraction of C undergoing chemical reaction

Concentration of the species in various subdomains Now that we have the necessary effective parameters for the homogenised cytoplasm subdomain, the diffusion equation for this domain can be set using these new parameters. Together with the diffusion equation of the other subdomains which were already homogeneous and straight forward right from the beginning, the diffusion problem of the whole domain can be solved for the concentrations C and U which are mean concentrations for each point in the cytoplasm subdomain. In the homogenised cytoplasm, we solve for C and U instead.

4.3.1

Reaction. Fraction of concentration affected by chemical reaction in the cytoplasm.

Generally, reaction is as described in Chapter 2.2, but in the homogenised cytoplasm, other considerations must be made. Since C is a weighted mean between the Cs’ in both Ωw and Ω f , and noting that in reality chemical reaction only takes place in the Ωw , we have to decide what fraction of C is involved in chemical reaction. In this case we have to apply the concept of adsorption isotherm[7, pp.407, 408]. An adsorption isotherm is an expression relating the quantity of an adsorbed quantity e.g in Ω f , to the quantity in another phase e.g in Ωw . We shall use the equilibrium isotherm, which states that the amount of adsorbed component is equal to the amount at equilibrium. This means that for any Ω f and its neighbouring Ωw , we assume that Cw = K pC f

(4.20)

This is a straight-forward isotherm, and is applicable when the phases Ω f and Ωw in a REV are tiny enough for almost instantaneous concentration equilibrium [7, pp.407, 408] with any one of the diffusion coefficients in our problem. The present problem is an example of this: This means that the equilibrium isotherm is very appropriate for the present problem. We know C = ρwCw + ρ f C f

(4.21)

Applying Equations 4.20, Equation 4.21 can be written C = ρw K pC f + ρ f C f = C f ρw K p + ρ f and hence 13




(4.22)

Cf =

C ρwCw + ρ f

(4.23)

and the concentration involved in chemical reaction is given by Cw = K pC f = K p

14

C ρwCw + ρ f

(4.24)

Chapter 5 Model Implementation with FEMLAB 5.1

The Femlab Software

FEMLAB is a software package for the simulation and visualisation of partial differential equations in one, two or three dimensions. The simulations are based on the finite element method, abbreviated as FEM, hence the name FEMLAB. It performs equation-based multiphysics modelling [13, p.153]. This means that we can formulate our equations so that they actually suit our problem. The physical domain is represented graphically in the software, and this is called the geometry, or computational domain. If various parts of the physical domain have different properties or phenomena, then the computational domain can be differentiated into subdomains which will have different parameters and, maybe, equations. The underlying mathematical structure of FEMLAB is a system of partial differential equations (PDE)s. There are many application modes in FEMLAB, suitable for various scientific problems. These are, so to speak, templates for defined equations which can be modified by changing the values of some predefined parameters, to suit the scientific problems we want to solve. The problem at hand does not fit well into the predefined application modes. Therefore the coefficient form of the PDE mode is used in this work. In this mode, one physics mode[10, p.8] can handle many variables. Since there are at least two variables in each part of the cell, this property is very useful for the present model.

5.2

The Geometry

A 2D model is sufficient for our purpose. A sector of only one-sixteenth (22.5 o ) of the cell is used to minimise computational resources and time. Thickness of the nucleus was used as the scaling factor. All parts of the cell are scaled with this factor s. 15

Since the thickness of the cytoplasm and nucleus are of the order of a 1000 times that of the membranes, the graphical representation of the membranes in the femlab Draw mode would be very thin. Therefore the triangular elements in these thin domains would be very tiny, and thus the model would be too computationally expensive. One way to solve this problem would be to approximate the membranes as simple boundaries, by a technique known as thin film approximation [4]. However, the example in the FEMLAB manual is straightforward and involves no partition coefficient, whereas in the present case, there is a partition coefficient because the membrane should act as a reservoir for A and U . Of course, this is possible, but rather inconvenient. Therefore, I chose a different approach. The extracellular solution, as will be shown below, is 27.748 times thicker than the nucleus whose scaled radius is 1: If we denote the scaled radius of the entire cell by r, and that of the medium surrounding it by R, then equating the ratio of their volumes to 168, we have R3 = 168 (5.1) r3 If we then substitute r by 4 in (5.1), we find R to be 22, and therefore the scaled thickness of the external solution alone is 22 − 4 = 18. Therefore our computational domain consists of: • A rectangle which represents the extracellular solution • A thin outer arc which represents the cell membrane • A thicker inner arc which represents the cytoplasm. The thousand of tiny membranes in the cytoplasm are not represented in the domain because they are handled by homogenisation. • A thin inner arc which represents the nuclear membrane • Finally, a thicker central circle which represents the nucleus All the arcs are concentric with the central circle and only 22.5o is taken, from the centre of the central circle, as seen in Figure 5.1. In the FEMLAB draw mode, the scaled domain in Figure 5.1 is drawn. The representation of a domain in FEMLAB is called a geometry [8, p.157].

5.3

Subdomain properties, equations and constants

From the multiphysics menu, a physics is chosen depending on which subdomain, and appropriate parameters are entered in the subdomain settings dialog box [8, p.157] to specify the diffusion-reaction equation in that subdomain. The coefficient form of the PDE mode is used. 16

Figure 5.1. The computational Domain

5.4

Constants and Parameters

It is most convenient to have all the variables and expressions defined in the options menu [10, p.98] . This is so that if we want to change any coefficients or material properties, we do not need to go to the subdomain or boundary settings and modify these for each subdomain. We just need to modify the value by going to the options menu. In order to know what the constants in my programme stand for, below is a table of them: 1

5.5

Subdomains and their properties

Subdomain properties of the appropriate physics are presented in Table 5.2. The boundaries are numbered anti-clockwise round each subdomain. We begin from boundary 11 , which is the left boundary of the extracellular water. The subscripts indicate the subdomains. Note that for simplicity, the boundary and sub-domain numbering in this report is not the same as in the programme. 1 The asterices (∗)in

Table 5.1 indicate given data. This is the data in the programme that may be changed by the experimenter. The rest of the entries in the table are computed by the programme.

17

constant conc cr cscale c0 Dext D1 D2 D2T D2P D3 D4 D parallel Dseries f rac1 f rac2 G Kc Kcc Ku M n N num pump pk pk2 Rcell Rw s sf pf S theta Tws Tw Vw Vn V1 Vratio wscale

meaning and units Initial concentration in the extracellular solution (M) ∗ Fraction of C undergoing chemical reaction Scaling factor for concentration * Normalised Initial concentration in the medium
Diffusion coefficient (D) in the extra-cellular solution
m2 s−1 ∗ D of the species in the outer membrane m2 s−1 ∗
Effective diffusion coefficient in the cytoplasm M · m2
s−1 Transverse D of the species in the membrane m2 s−1
∗ Normal D of the species in the membrane m2 s−1
∗ D of the species in the nuclear membrane m
2 s−1 ∗ D of the species in the nucleus −2 s−1 ∗
D in homogenised cytoplasm if in parallel m2 s−1
D in homogenised cytoplasm if in series m2 s−1 Volume fraction of cytoplasm occupied by lipid part * Volume fraction of occupied by acquous part Concentration of GST (M)
∗ Catalytic activity M −1 · s−1 ∗
Reaction constant of C with GST in cytoplasm =
G · Kc s−1 Reaction constant of C with water s−1 ∗ Mass transfer coefficient * Number of cells whose volume was measured as Vn ∗ Number of cells in culture * s f + sp factor determining the rate at which B is pumped out ∗ Partition coefficient between aqueous and lipid parts ∗ Partition coefficient between membrane and homogenised cytoplasm Radius of cell (m) Scaled radius of part of medium containing cell (m) Radius of nucleus. Space scaling factor for cell (m) ∗ Portion of membranes in cytoplasm parallel to diffusion path Portion of membranes in cytoplasm normal to diffusion path Scaling factor of extracellular space in addition to s (m) Angle of sector of circle representing the cell (radians) Scaled hickness of part of medium enclosing one cell (m) Thickness of part of medium enclosing one cell (m)
Volume of part medium enclosing
one cell m3 Volume of n cells m3 ∗
Volume of one cell m3 Volume ratio of medium per cell Total scaling factor for extracellular space (m) Table 5.1. Constants used in the programme

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value 1 × 10 −4 347 × 10−6 1 × 10−4 1.0 1.3 × 10−11 1 × 10−12 3.9517 × 10−11 1 × 10−10 1 × 10−12 1 × 10−12 1 × 10−14 5.926 × 10−11 2.419 × 10−14 0.592593 1 − 0.592593 347 × 10−6 660003 2.244 3.6 × 10−4 1 × 10−4 1 × 107 2 × 107 3 0.02 1 × 10−4 1 × 10−4 1.92 × 10−4 Computed by code 4.8 × 10−6 2 1 55.496 π 8

Computed by code Computed by code 1 × 10−5 3 × 10−6 Computed by code Computed by code 1.3874 × 10−4

Physical subdomain Femlab Subdomain Variables Water 1 C1 , U1 Cellmembrane 2 C2 , U2 Cytoplasm 3 C3 , B3 , U3 Nuclearmembrane 4 C4 , U4 Nucleus 5 C5 , A5 , U5

boundaries 11 , 21 ,31 , 41 32 , 52 ,62 , 72 63 , 83 ,93 , 103 94 , 114 ,124 , 134 125 , 145 ,155

Table 5.2. Subdomains

5.6

Initial condition

In the sub-domamain settings mode, all initial concentrations were left at their default value, which is 0, except that the scaled initial concentration of C in subdomain 1 was changed to c0 .

5.7

Boundary conditions

In the boundary settings dialog box [12] all the exterior boundaries of the domain were left at their default (insulation). Then choosing the appropriate physics in FEMLAB, the default insulation was again left unchanged for species which do not diffuse out of their subdomains. These are A and B. Then for C and U , the flux was set according to (2.10). This is similar to separation through dialysis model of the FEMLAB model library [3, p.213]. While all boundaries are insulated for A and B, table 3 below sumarises the boundary conditions for C and U . Boundary number subscripts 3, 6, 9, 12 remaining numbers

Type flux insulation

Table 5.3. Boundary conditions for C and U

5.8

Solving

The problem was solved with the default solver parameters. It was sufficient to solve the problem up to 250 seconds. Solution time was only 1 minute, due to the efficiency of the programme thanks to homogenisation. 19

5.9

Results

In Figure 5.2, concentrations inside the cytoplasm with time, are compared to the in vitro results. The concentrations from the in vitro experiments were taken only at a limited number of instants because it is a very difficult task. The scanty dots represent concentrations from in vitro experiments, while the graphs are from the simulations. It is not the actual concentrations which are plotted here, but the percentage, where the maximum is set to 100 %. Overall, the patterns are similar. The differences seen might be explained by that the molecular dynamics within the cell is more complex than we assume in our model.

Figure 5.2. C, B and U in cytoplasm plotted with time

Plots of the concentration of C in the extracellular solution have been compared to similar plots obtained experimentally by the researchers at K.I. It shows the half life of C. Please see Figure 5.3. Laboratory experiments showed that the half life of the substrate C in the solution was about 63 seconds. The plot in Figure 5.2 obtained from the FEMLAB simulation depicts a half-life of 60 seconds. This is close enough.

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Figure 5.3. C in solution plotted against time

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Chapter 6 Conclusion In the laboratory, careful and tedious measurement techniques are necessary in order to know the concentration of, for instance, A in the nucleus. Since the model reproduces those obtained in the laboratory, it can therefore be used as a quick and easy alternative to determine how much of the carcinogenic product A is present in the nucleus, which is an indication of the risk of cancer. Here is a plot of the concentration of A in the nucleus for 500 seconds. One can conclude that the model can fulfil its aims as indicated in Chapter 1. The results encourage us to continue developing the model alongside in vitro experiments.In this model, the chemical reactions occur within the bulk of the aqueous compartment of the cytoplasm. In future, a similar model is intended to be used to solve problems where surface reactions are also included. This is the case where the enzyme which is only present on the surface of the membranes. The present model will, of course, facilitate the modelling of surface reaction problems, but the homogenisation of surface reactions are very complex, and a deeper understanding of the physics and mathematics involved in transport phenomena is needed.

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Figure 6.1. Concentration of A in the nucleus

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References 1. K. Sundberg, K. Dreij, A. Seidel and B. Jernstrom.Gluthion Conjugation and DNA Adduct Formation of Dibenzol [a, 1] pyrene and Benzo [a] pyrene Diol Epoxides in V79 Cells Stably expressing Different Human Glutathione Transferases. Chemical research in Toxicology ,2002,15, pp. 170-179 2. A. Hartmann, E. M. Golet, S. Gartiser, et al. Primary DNA Damage But Not Mutagenicity Correlates with Ciprofloxacin Concentrations in German Hospital Wastewaters Archives of Environmental Contamination and Toxicology, February 1999, pp. 115 - 119. 3. COMSOL AB, Chemical Engineering Module. 2004, Stockholm, p. 213 4. COMSOL AB, Thin Film Approximation, http://www.comsol.com/support/knowledgebase/902.php. 5. H. Fredrick, A. Barbero. Steady-state flux and lag time in stratum cornium lipid pathway using finite element methods. Journal of pharmaceutical Sciences. Vol 92, NO.11, November 2003, pp. 2196-2207. 6. Lars Erik Persson, Leif Persson, Nols Svanstedt and John Wyller, The Homogenization Method, an introduction. England, Chartwell Bratt, 1993. 7. Jacob Bear and Yehuda Bachmat, Introduction to Modelling of Transport Phenomena in Porous Media, The Netherlands, Kluwer Academic , 1990. 8. COMSOL AB, Model Library, FEMLAB3. Stockholm, 2004. 9. COMSOL AB, Chemical Engineering Module, Stockholm, 2002, pp. 2-253, 2-254. 10. COMSOL AB, FEMLAB3 Users’ Guide. Stockholm, 2004.

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List of Abbreviations A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DNA adducts B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . gluthione conjugates b.c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . boundary condition D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . diffusion coefficient Dar . . . . . . . . . . . . . . . . . . . . . . weighted arithmetic mean diffusion coefficient Deff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . effective diffusion coefficient Dha . . . . . . . . . . . . . . . . . . . . . . weighted harmonic mean diffusion coefficient C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . diol epoxides GST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . gluthatione transferase K.I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Karolinska Institutet K p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . partition coefficient REV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . representative elementary volume U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . tetrols

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