FEMLAB - KTH
FEMLAB - KTH
FEMLAB - KTH
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Mathematical Modelling<br />
and Numerical Solution of Chemical<br />
Reactions and Diffusion of Carcinogenic<br />
Compounds in Cells<br />
DONALD O BESONG<br />
Master’s Degree Project<br />
Stockholm, Sweden 2004<br />
TRITA-NA-E04152
Numerisk analys och datalogi Department of Numerical Analysis<br />
<strong>KTH</strong> and Computer Science<br />
100 44 Stockholm Royal Institute of Technology<br />
SE-100 44 Stockholm, Sweden<br />
Mathematical Modelling<br />
and Numerical Solution of Chemical Reactions<br />
and Diffusion of Carcinogenic Compounds in Cells<br />
DONALD O BESONG<br />
TRITA-NA-E04152<br />
Master’s Thesis in Numerical Analysis (20 credits)<br />
at the Scientific Computing International Master Program,<br />
Royal Institute of Technology year 2004<br />
Supervisor at Nada was Michael Hanke<br />
Examiner was Axel Ruhe
Abstract<br />
In order to shed more light on how cancer is triggered, Professor Bengt Jernstrom<br />
and his research group at Karolinska Institute (KI) have been performing<br />
in vitro incubation of carcinogenic compounds with cells. In vitro reactions and<br />
diffusion take place when the carcinogenic substrate is added to cells in culture.<br />
Only one cell and its appropriate quota of the medium is needed for the mathematical<br />
model, and indeed only a 22.5 o sector of a cell is modelled. <strong>FEMLAB</strong> is<br />
the software used for the simulation. The graphical representation of the problem<br />
and its simulation is made possible by applying the mathematical technique of homogenisation<br />
in the multi-compartment cytoplasm. All constants and parameters<br />
used in the simulation were the same used for the in vitro experiments. The model,<br />
and consequently the programme, can be adapted to various physical and chemical<br />
scenarios.<br />
The concentration of the carcinogenic substrate in the extracellular solution is<br />
computed, and its half-life is compared to the in vitro results. Both results are found<br />
to be the same.<br />
The model can be used for the prediction of the experimental inaccessible concentration<br />
profile in the nucleus.<br />
Matematisk modellering och numerisk lösning av<br />
reaktioner och diffusion för cancerogena ämnen i<br />
celler<br />
Sammanfattning<br />
För att belysa hur cancer uppkommer, har prof Bengt Jernström och hans forskargrupp<br />
p˚a Karolinska Institutet (KI) utfört in vitro odling av cancerogena ämnen<br />
i celler, där reaktioner och diffusion d˚a äger rum. Endast en cell behövs för att<br />
sätta upp en matematisk modell, och av denna cell modelleras endast en 22.5<br />
graders sektor. <strong>FEMLAB</strong> har använts för simuleringen. Den grafiska representationen<br />
av problemsimuleringen har möjliggjorts genom att applicera homogenisering
p˚a multi-compartment cytoplasma. Alla konstanter och parametrar som använts i<br />
modellen hade samma värden som i in vitro experimenten. Modellen, och även programmet,<br />
kan anpassas till olika fysikaliska och kemiska scenarier. Koncentrationen<br />
av de cancerogena ämnena i modellen och deras halvtids livslängder beräknas<br />
och jämförs med in vitro resultat. B˚ada resultaten överensstämmer. Modellen kan<br />
användas för prediktion av omätbara koncentrationer i cellkärnan.
Contents<br />
1 Introduction 1<br />
2 The Physical Problem and its Mathematical Model 3<br />
2.1 Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4<br />
2.2 Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4<br />
2.3 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 5<br />
2.4 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . 5<br />
3 Scaling and Reformulation 7<br />
3.1 The diffusion reaction model equation . . . . . . . . . . . . . . . 8<br />
4 Simplification of problem by means of homogenisation 9<br />
4.1 Finding D eff . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10<br />
4.1.1 Weighted arithmetic mean of the diffusion coefficient . . . 10<br />
4.1.2 Weighted harmonic mean diffusion coefficient . . . . . . 11<br />
4.1.3 Decision on D eff for the cytoplasm . . . . . . . . . . . . 12<br />
4.2 Partition coefficient between homogenised cytoplasm and other subdomains<br />
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12<br />
4.3 Solving for concentration; Fraction of C undergoing chemical reaction<br />
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13<br />
4.3.1 Reaction. Fraction of concentration affected by chemical<br />
reaction in the cytoplasm. . . . . . . . . . . . . . . . . . 13<br />
5 Model Implementation with <strong>FEMLAB</strong> 15<br />
5.1 The Femlab Software . . . . . . . . . . . . . . . . . . . . . . . . 15<br />
5.2 The Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15<br />
5.3 Subdomain properties, equations and constants . . . . . . . . . . 16<br />
5.4 Constants and Parameters . . . . . . . . . . . . . . . . . . . . . . 17<br />
5.5 Subdomains and their properties . . . . . . . . . . . . . . . . . . 17<br />
5.6 Initial condition . . . . . . . . . . . . . . . . . . . . . . . . . . . 19<br />
5.7 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . 19<br />
5.8 Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.9 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20<br />
6 Conclusion 22<br />
References 24<br />
List of Abbreviations 25
Chapter 1<br />
Introduction<br />
The aim of this work is to develop a mathematical model for the in vitro chemical<br />
reactions and diffusion of carcinogenic compounds in cells, and then create a computer<br />
programme based on the model. Consequently, by using different parameters<br />
the programme will enable us to carry out these experiments virtually, and thus<br />
predict the risk of cancer in living humans and animals.<br />
The primary carcinogenic substrate, in the form of diol epoxides (C), is initially<br />
outside the cell. Outside the cell, some of this substrate reacts with water to form<br />
tetrols (U), which do not cause cancer, although it diffuses everywhere in the cell.<br />
When the remaining substrate C diffuses through the outer membrane of the cell,<br />
it still reacts with water within the cytoplasm, while some of it is converted into<br />
glutathione conjugates (B) by an enzyme called GST . B does not cause cancer<br />
either, but remains in the cytoplasm, where it is pumped out. The remaining C will<br />
reach the nucleus, where it still reacts with water in the nucleus to form U, as well<br />
as with DNA to form DNA adducts (A). It is this A that causes cancer. It will stay<br />
in the nucleus without diffusing into the other parts of the cell.<br />
Each cell in the cell culture used in the study is surrounded by about 168 times<br />
its volume of medium. The substrate was then added in the medium for the reactions<br />
and diffusion to begin.<br />
There are no reactions in the membranes, or so-called lipid compartment of<br />
the cell. Reaction only takes place in the aqueous compartment of the cell: In the<br />
cytoplasm the reaction of substrate C with water is slow but the enzymatic reaction<br />
is much faster. In the nucleus, the reaction of C with DNA is the same rate as the<br />
reaction with water in the cytoplasm.<br />
Figure 1.1 illustrates the diffusion and reactions in and out of a cell.<br />
The partition coefficient, Kp, which is the equilibrium ratio of the concentration<br />
C or U between any aqueous compartment of the cell and its neighbouring lipid<br />
compartment, is 1 × 10 −3 . Kp depends on the substrate.<br />
The complexity of this work lies on the complex geometry, as well as the fact<br />
1
Figure 1.1. Illustration of reactions and diffusion in a cell<br />
that there are many reactions with different products, and their diffusion, in each<br />
part of the cell. With mass transfer between various parts of the cell, the work has<br />
proved to be more complex, but the bottle-neck is a geometry with heavily varying<br />
details. This is handled by applying scaling and homogenisation. We shall go into<br />
this after studying the physical domain as presented in chapter 2, which reveals<br />
the cytoplasm as consisting of thousands of interconnected tiny parts of the lipid<br />
compartment sandwiched between very tiny parts of the aqueous compartment.<br />
Chapter 2 sets the physical problem and formulates its mathematical model. It<br />
is most important to work with a dimensionless model, so in Chapter 3, the model<br />
is scaled and reformulated. In Chapter 4, the homogenisation of the cytoplasm is<br />
explained, as well as how the solution of the modified problem is obtained with<br />
a knowledge of equilibrium isotherms [7, pp.407,408]. A brief presentation of the<br />
<strong>FEMLAB</strong> software is found in Chapter 5, where the implementation of the model is<br />
described. Also in Chapter 5, there is a list of the variable names in the programme,<br />
with their physical meanings, so that users of the programme may easily adapt it to<br />
their own needs.<br />
The in vitro experiments were performed by Kristian Dreij, who is presently a<br />
PhD student of toxicology under the supervisors prof. Bengt Jernstrom and prof.<br />
Ralf Morgenstern at K.I. They all, as well as my supervisor at KT H doc. Michael<br />
Hanke, have always been quick to help me in this work.<br />
2
Chapter 2<br />
The Physical Problem and its<br />
Mathematical Model<br />
The concentration of the cells in the culture is one cell per 168 times its volume<br />
of the medium. Therefore, the smallest useful part of the physical domain for this<br />
study is a cell surrounded by about 168 times its volume of the substrate solution.<br />
Originally, the substrate C is found only in this extra-cellular solution.<br />
The cell consists of the cell membrane which has a thickness of 0.16 percent of<br />
the radius of the nucleus. Beyond the cell membrane is the cytoplasm which has<br />
a thickness of 3 times the radius of the nucleus, and the nuclear membrane which<br />
has the same thickness as the cell membrane. At the centre of the cell we have the<br />
nucleus, 4.8 × 10 −6 m in radius.<br />
We shall call the outer and nuclear membranes, together with the inner membranes<br />
within the cytoplasm, the lipid compartment of the cell, and the rest of the<br />
cell the aqueous compartment, which comprises both the cytoplasm and nucleus<br />
minus any membranes therein. Note that this aqueous compartment of the cell is<br />
not the same as the extracellular water.<br />
The cytoplasm contains thousands of inter-connected membrane-like sheets,<br />
as well as tiny structures consisting of membranes of the same thickness as the<br />
outer membrane. Examples of these structures are the endoplasmic reticulii and the<br />
mitochondria. Some of these structures are closed and oval in shape, but contain<br />
a aqueous compartment. The nucleus is free from any membranes. However, the<br />
ratio of all the lipid compartment of the cell to the volume of the whole cell is<br />
about 25 : 100. That is, one quarter of the cell is made up of these membranes.<br />
Moreover, since the scaled radius of the whole cell is 4, the scaled thickness of the<br />
cytoplasm 3, and the radius of the nucleus 1, and all these membranes are found in<br />
the cytoplasm except the cell membrane and the nuclear membrane which can be<br />
neglected, the volume fraction of the lipid compartment in the cytoplasm is given<br />
3
y<br />
Figure 2.1. Schematic picture of the cell showing cytoplasm and nucleus<br />
�<br />
43 0.25<br />
43 − 13 �<br />
= 0.254 (2.1)<br />
The cytoplasm is therefore packed with membranes, as can be seen in Figure<br />
2.1.<br />
This is a diffusion/reaction problem. There is no advection. I will first describe<br />
the diffusion modelling and then the reaction modelling. Finally, I will combine<br />
them.<br />
2.1 Diffusion<br />
Let us call the ith subdomain Ωi, and let us assume that the concentration of the<br />
diffusing species in Ωi is φ. Let the diffusion coefficient of φ in Ωi be D. Then, the<br />
rate of change of φ in Ωi is given by<br />
2.2 Reaction<br />
∂φ<br />
∂t = D ▽2 φ (2.2)<br />
In this study, C is the only compound which reacts. Let the concentration of C, B, U<br />
and A in Ωi be denoted by Ci, BiUi and AiCi respectively. There are three reactions:<br />
4
Ci + water −→ Ui at rate kiCi<br />
(2.3)<br />
where ki is the rate constant of the reaction with water in the ith subdomain subdomain.<br />
Ci + GST −→ Bi at rate kBiCi · GST (2.4)<br />
where the enzyme GST has so many active sites that the reaction rate is always<br />
constant.<br />
Ci + DNA −→ Ai at rate kAiCi (2.5)<br />
kAi and kBi are the rate constants of the reaction with A and B respectively in the<br />
ith subdomain subdomain. If a certain reaction does not take place in some subdomain<br />
Ωi, then the appropriate reaction rate is simply 0. For example, there are no<br />
reactions in the lipid compartment of the cell (membranes), so all the reaction rates<br />
there are 0.<br />
For instance, in the nucleus Ωnu, the models for the reactions represented by<br />
(2.3) and (2.5) are:<br />
dCnu<br />
dt = −(knu + kAnu)Cnu (2.6)<br />
where the subscript nu indicates that the variable belongs to the nucleus.<br />
2.3 Initial Conditions<br />
dUnu<br />
dt = +knuCnu (2.7)<br />
dAnu<br />
dt = +kAnuCnu (2.8)<br />
In all the subdomains, the initial concentration is 0 for all the substances C,U,B,<br />
and A, except that the initial concentration of C in the rectangular domain Ω0 representing<br />
the extracellular solution, is C0.<br />
2.4 Boundary Conditions<br />
On the left boundary of the rectangular subdomain we have a homogeneous boundary<br />
condition because of symmetry. On its upper and lower boundaries we also<br />
have homogeneous boundary condition because there are no concentration gradients<br />
along their normals. The upper and lower boundaries of the sector-like geometry<br />
to the right of the rectangular subdomain we also have zero Neumann boundary<br />
5
conditions because there are no concentration gradients in the direction of their<br />
normals. Hence, for any species φi at those boundaries, the flux is given by:<br />
n · (D ▽ φi) = 0 (2.9)<br />
where n is the normal to the boundary.<br />
So far, I have mentioned only external boundaries. I shall consider internal<br />
boundaries now: The first internal boundary from the left is that between the rectangular<br />
subdomain and the curved boundary. In reality, the whole domain is continuous<br />
here, but I decided to split them because of the different scaling factors<br />
applied in the rectangular subdomain in the radial direction. To avoid polar coordinates,<br />
and taking advantage of the fact that there are no concentration gradients<br />
in the θ-direction, I have straightened that subdomain into a rectangle, as opposed<br />
to its original arc shape. Then I introduced coupling variables between its right<br />
boundary and the left boundary of the remaining part of the geometry.<br />
Generalising, if a certain species φi only stays within a certain subdomain Ωi<br />
and does not diffuse through, then (2.9) holds for that species. However, if the<br />
species diffuses through the boundary separating say Ω1 and Ω2, then the flux will<br />
be a function of φ1 and φ2 at the boundary. Moreover, if the partition coefficient for<br />
φ is Kp between Ω1 and Ω2, where Kp < 1, then the flux into Ω1 is given by<br />
n · D ▽ φi = M (φ2 − Kpφ1) (2.10)<br />
and that into Ω2 through that boundary is simply the negative of the flux into Ω1. M<br />
is the mass transfer coefficient, and it is a measure of the resistance to the transport<br />
of any given species between the two given subdomains. A high M implies a small<br />
resistance to mass transfer, and vice-versa.<br />
6
Chapter 3<br />
Scaling and Reformulation<br />
The scaled domain is seen in Figure 3.1 below.<br />
The scaling was done as follows:<br />
Figure 3.1. The computational Domain<br />
˜x = x<br />
S1<br />
˜y = y<br />
S1<br />
7<br />
(3.1)<br />
(3.2)
where ( ˜x, ˜y) are the new coordinates of the computational domain, and S is the<br />
scaling factor. For all sub-domains within the cell itself, the scaling factor S1 =<br />
2.24 −6 is the radius of the nucleus. For the rectangular domain, scaling by only S1<br />
makes its radius to be 18, which is more than four times the radius of the whole cell.<br />
To make it graphically convenient, it is again scaled by yet another scaling factor,<br />
in order to reduce its thickness to 0.5. Therefore, for the rectangular sub-domain in<br />
Figure 3.1, the scaling factor is S2 = 36 × S1.<br />
The concentration of any given species are also scaled:<br />
˜φi = φi<br />
C0<br />
where C0 is the initial concentration of C in the water surrounding the cell.<br />
3.1 The diffusion reaction model equation<br />
(3.3)<br />
Applying the above scaling, and combining diffusion and reaction, the general<br />
equation for any given subdomain is given by<br />
∂ ˜φi<br />
∂t<br />
= D<br />
S ▽2 ˜φi + F˜φi<br />
(3.4)<br />
where ˜φ is any of the scaled species, S = S1 or S2 the scaling factor for space, and F<br />
is the reaction term representing the rate of change of the scaled concentration due<br />
to reaction in that subdomain. Let us again take the example of the nucleus. Then<br />
φi is given by<br />
∂φnu ˜ D<br />
=<br />
∂t S2 ▽<br />
1<br />
2 ˜φnu + F˜φnu<br />
(3.5)<br />
where φ is any of the species’ concentration present in the nucleus, namely C, U or<br />
A, and F is the right-hand side of equations (2.6), (2.7), or (2.8), depending on the<br />
appropriate species, and the tilde sign simply means it is normalised by the scaling<br />
factor C0.<br />
8
Chapter 4<br />
Simplification of problem by means of<br />
homogenisation<br />
The computational domain in Figure 5.1 represents a cell whose cytoplasm is homogeneous,<br />
rather than one which has many tiny membranes. Therefore, making<br />
the cytoplasm homogeneous would be a wise idea. This method is know as homogenisation.<br />
In reality, the cytoplasm is extremely densely packed with lipophilic compartments,<br />
in the form of endoplasmic reticulii, mitochondria, etc. Such a set-up is<br />
similar to a porous medium [7, p.5]. Therefore, the cytoplasm is a multicompartment<br />
medium Ω consisting of two componets: the lipophilic or fatty domain Ω f<br />
and the hydrophilic or watery domain Ωw. We then assume that the mixture is homogeneous,<br />
i.e. a representative elementary volume (REV) taken anywhere in Ω<br />
is identical. Let<br />
ρ f = total volume of Ω f<br />
total volume of Ω<br />
(4.1)<br />
and<br />
total volume of Ωw<br />
ρw = 1 − ρ f =<br />
total volume of Ω<br />
(4.2)<br />
Then the following steps are taken in order to model what happens in the cytoplasm:<br />
• finding an effective diffusion coefficient � �<br />
Deff for the homogenised cytoplasm<br />
• finding a new partition coefficient between the other parts of the cell and the<br />
homogenised cytoplasm<br />
• using the above to get the concentration of any diffusing species φ for every<br />
point in the homogenised cytoplasm.<br />
9
• applying an appropriate isotherm [7, pp.407,408] to get the part of the concentration<br />
involved in reaction, since only the C within the aqueous compartment<br />
reacts, remembering that no reaction takes place in the lipid compartments<br />
or membranes.<br />
4.1 Finding D eff<br />
The diffusion path in this model is radial, directed from the cell membrane to the<br />
nucleus. If all the membranes in the cytoplasm were oriented parallel to the diffusion<br />
path, then D eff = Dar, where Dar is the weighted arithmetic mean diffusion<br />
coefficient. If all the membranes in the cytoplasm were oriented perpendicular to<br />
the diffusion path, then D eff = D ha , where D ha is the weighted harmonic mean<br />
diffusion coefficient. In principle, in any intermediate case, Dar ≤ D eff ≤ D ha or<br />
Dar ≥ D eff ≥ D ha [6, p.10].<br />
4.1.1 Weighted arithmetic mean of the diffusion coefficient<br />
The weighted arithmetic mean is obtained if all the Ω f subdomains are oriented<br />
parallel to the diffusion path. That is, they are radially oriented in the cytoplasm. If<br />
we take a REV containing one Ω f and one Ωw subdomain, then the rate of change<br />
of the concentration of any species φ is given by<br />
∂φ<br />
∂t = ∂φ (w)<br />
∂t + ∂φ ( f )<br />
∂t<br />
(4.3)<br />
where the subscripts w and f denote the watery and fatty subdomains, respectively.<br />
If the total volume of this REV is V , we can derive an average concentration for<br />
it as follows:<br />
φ = 1<br />
Z<br />
φdv =<br />
v v<br />
1<br />
Z<br />
φ<br />
v<br />
(w)dv +<br />
v<br />
1<br />
Z<br />
φ ( f<br />
v<br />
)dv (4.4)<br />
v<br />
But since φ (w) is non-zero only in Ωw, and φ ( f ) non-zero only in Ω f , (4.4) becomes<br />
∂φ<br />
∂t<br />
Z<br />
1<br />
=<br />
v v<br />
Z<br />
∂φ 1 ∂φ (w)<br />
dv =<br />
∂t v vw ∂t dvw + 1<br />
Z<br />
v v f<br />
∂φ ( f )<br />
∂t dv f (4.5)<br />
where vw and v f are the volumes of the Ωw and Ω f subdomains respectively. But<br />
the rate of change of concentration is given by (2.2). Therefore if the average<br />
quantity ▽φ is ▽φ, and hence the average quantity ▽ 2 φ is ▽ 2 φ, then (4.5) becomes<br />
∂φ<br />
∂t<br />
Z<br />
1<br />
= D<br />
v eff▽ v<br />
2φdv = 1<br />
Z<br />
Dw▽<br />
v vw<br />
2φdvw + 1<br />
Z<br />
D f ▽<br />
v v f<br />
2φdv f<br />
10<br />
(4.6)
and since ▽ 2 φ is constant over the REV just mentioned,<br />
∂φ<br />
∂t<br />
1<br />
=<br />
v Deff▽2φ·v = 1<br />
v Dw▽2φ·vw + 1<br />
v D f ▽2φ·v f = ▽2 �<br />
1<br />
φ·<br />
v Dw · vw + 1<br />
v D �<br />
f · v f<br />
Applying (4.1) and (4.2), equation (4.7) becomes<br />
and finally<br />
∂φ<br />
∂t = Deff▽2φ = ▽2φ · � �<br />
ρwDw + ρ f D f<br />
D eff = ρwDw + ρ f D f = Dar<br />
4.1.2 Weighted harmonic mean diffusion coefficient<br />
(4.7)<br />
(4.8)<br />
(4.9)<br />
The weighted harmonic mean is obtained if all the Ω f subdomains are oriented<br />
normal to the diffusion path. That is, they are in series with the Ωw subdomains. If<br />
we take a REV containing one Ω f and one Ωw subdomain, then the rate of change<br />
of the average concentration of any species φ is given by<br />
∂φ<br />
∂t = D · ▽2 φ (4.10)<br />
In this case in series, ▽ 2 φ is considered separate for Ωw and Ω f . i.e<br />
Also,<br />
and<br />
where ∂φ<br />
∂t is for both Ωw and Ω f .<br />
or<br />
▽ 2 φ = ▽ 2 φ w + ▽ 2 φ f<br />
▽2φw = D −1 ∂φ<br />
w ·<br />
∂t<br />
▽ 2 φ f = D −1<br />
f · ∂φ<br />
∂t<br />
(4.11)<br />
(4.12)<br />
(4.13)<br />
Combining (4.10), (4.11), (4.12) and (4.13),<br />
Z<br />
1<br />
�<br />
▽<br />
v v<br />
2φw + ▽2 �<br />
φ f dv = ∂φ<br />
Z<br />
1<br />
D<br />
∂t v vw<br />
−1<br />
w dvw + ∂φ<br />
Z<br />
1<br />
D<br />
∂t v v f<br />
−1<br />
f dv f (4.14)<br />
Therefore<br />
or<br />
▽2φ = ∂φ<br />
�<br />
ρw · D<br />
∂t<br />
−1<br />
w + ρ f · D −1<br />
�<br />
f<br />
D −1<br />
eff = ρw · D −1<br />
w + ρ f · D −1<br />
f<br />
�<br />
Deff = ρw · D −1<br />
w + ρ f · D −1<br />
�−1 f = Dha 11<br />
(4.15)<br />
(4.16)<br />
(4.17)
4.1.3 Decision on D eff for the cytoplasm<br />
The thin sheet-like membranes in the cytoplasm may take any orientation. Let us<br />
now consider the 3D cell to have its membranes in any one of three orientations:<br />
perpendicular to the diffusion path, radially oriented but vertical, or radially oriented<br />
but horizontal. This is a perfect, un-biased orientation of the membranes,<br />
with one-third of the membranes in each direction. Therefore one-third of the<br />
membranes are in series with the aqueous compartment of the cytoplasm, while<br />
two-thirds is in parallel, with respect to the diffusion path. Therefore<br />
Deff = a · Dha + b · Dar<br />
a + b<br />
(4.18)<br />
where a = 1 and b = 2. Therefore, depending on what fraction of the membranes we<br />
think are oriented in each direction vis-a-vis the three above-mentioned directions,<br />
a and b can be changed. However, I and the professors have thought that the unbiased<br />
mode of orientation is most natural. Magnified 3D electron micrographs of<br />
the cell depict such an orientation of the membranes.<br />
Note that the diffusion coefficient along a membrane is not the same as across<br />
it. Therefore, in calculating D ha , a different D is used than when calculating Dar.<br />
This is implemented in the application.<br />
4.2 Partition coefficient between homogenised cytoplasm<br />
and other subdomains<br />
Now that the cytoplasm has been homogenised, all its physical properties have<br />
changed. We know the partition coefficient for the species C and U, between Ωw<br />
and Ω f is Kp < 1. This implies that at equilibrium, Cw = Kp ·Cf and Uw = Kp ·Uf .<br />
In other words, the concentration of either of those species in Ω f is K−1 p times<br />
greater than in Ωw.<br />
We now have to remember that the neighbouring subdomains to the homogeneous<br />
cytoplasm are in Ω f : viz the outer membrane and the nuclear membrane. If<br />
we denote the partition coefficient between this homogenised cytoplasm and any<br />
Ω f by Kp, ˆ then Kp ˆ can be derived. With little arithmetics, we have<br />
Kp ˆ = 1 · ρw + K−1 p · ρ f<br />
12<br />
K −1<br />
p<br />
(4.19)
4.3 Solving for concentration; Fraction of C undergoing<br />
chemical reaction<br />
Concentration of the species in various subdomains<br />
Now that we have the necessary effective parameters for the homogenised cytoplasm<br />
subdomain, the diffusion equation for this domain can be set using these<br />
new parameters. Together with the diffusion equation of the other subdomains<br />
which were already homogeneous and straight forward right from the beginning,<br />
the diffusion problem of the whole domain can be solved for the concentrations C<br />
and U which are mean concentrations for each point in the cytoplasm subdomain.<br />
In the homogenised cytoplasm, we solve for C and U instead.<br />
4.3.1 Reaction. Fraction of concentration affected by chemical<br />
reaction in the cytoplasm.<br />
Generally, reaction is as described in Chapter 2.2, but in the homogenised cytoplasm,<br />
other considerations must be made. Since C is a weighted mean between the<br />
Cs’ in both Ωw and Ω f , and noting that in reality chemical reaction only takes place<br />
in the Ωw, we have to decide what fraction of C is involved in chemical reaction. In<br />
this case we have to apply the concept of adsorption isotherm[7, pp.407,408]. An<br />
adsorption isotherm is an expression relating the quantity of an adsorbed quantity<br />
e.g in Ω f , to the quantity in another phase e.g in Ωw .<br />
We shall use the equilibrium isotherm, which states that the amount of adsorbed<br />
component is equal to the amount at equilibrium. This means that for any Ω f and<br />
its neighbouring Ωw, we assume that<br />
Cw = KpCf<br />
(4.20)<br />
This is a straight-forward isotherm, and is applicable when the phases Ω f and<br />
Ωw in a REV are tiny enough for almost instantaneous concentration equilibrium<br />
[7, pp.407,408] with any one of the diffusion coefficients in our problem.<br />
The present problem is an example of this: This means that the equilibrium<br />
isotherm is very appropriate for the present problem.<br />
We know<br />
C = ρwCw + ρ fCf<br />
Applying Equations 4.20, Equation 4.21 can be written<br />
� �<br />
C = ρwKpCf + ρ fCf = Cf ρwKp + ρ f<br />
and hence<br />
13<br />
(4.21)<br />
(4.22)
Cf =<br />
C<br />
ρwCw + ρ f<br />
and the concentration involved in chemical reaction is given by<br />
C<br />
Cw = KpCf = Kp<br />
ρwCw + ρ f<br />
14<br />
(4.23)<br />
(4.24)
Chapter 5<br />
Model Implementation with <strong>FEMLAB</strong><br />
5.1 The Femlab Software<br />
<strong>FEMLAB</strong> is a software package for the simulation and visualisation of partial differential<br />
equations in one, two or three dimensions. The simulations are based<br />
on the finite element method, abbreviated as FEM, hence the name <strong>FEMLAB</strong>. It<br />
performs equation-based multiphysics modelling [13, p.153]. This means that we<br />
can formulate our equations so that they actually suit our problem. The physical<br />
domain is represented graphically in the software, and this is called the geometry,<br />
or computational domain. If various parts of the physical domain have different<br />
properties or phenomena, then the computational domain can be differentiated into<br />
subdomains which will have different parameters and, maybe, equations.<br />
The underlying mathematical structure of <strong>FEMLAB</strong> is a system of partial differential<br />
equations (PDE)s. There are many application modes in <strong>FEMLAB</strong>, suitable<br />
for various scientific problems. These are, so to speak, templates for defined equations<br />
which can be modified by changing the values of some predefined parameters,<br />
to suit the scientific problems we want to solve.<br />
The problem at hand does not fit well into the predefined application modes.<br />
Therefore the coefficient form of the PDE mode is used in this work. In this mode,<br />
one physics mode[10, p.8] can handle many variables. Since there are at least two<br />
variables in each part of the cell, this property is very useful for the present model.<br />
5.2 The Geometry<br />
A 2D model is sufficient for our purpose. A sector of only one-sixteenth (22.5 o ) of<br />
the cell is used to minimise computational resources and time.<br />
Thickness of the nucleus was used as the scaling factor. All parts of the cell are<br />
scaled with this factor s.<br />
15
Since the thickness of the cytoplasm and nucleus are of the order of a 1000<br />
times that of the membranes, the graphical representation of the membranes in the<br />
femlab Draw mode would be very thin. Therefore the triangular elements in these<br />
thin domains would be very tiny, and thus the model would be too computationally<br />
expensive.<br />
One way to solve this problem would be to approximate the membranes as<br />
simple boundaries, by a technique known as thin film approximation [4]. However,<br />
the example in the <strong>FEMLAB</strong> manual is straightforward and involves no partition<br />
coefficient, whereas in the present case, there is a partition coefficient because the<br />
membrane should act as a reservoir for A and U. Of course, this is possible, but<br />
rather inconvenient. Therefore, I chose a different approach.<br />
The extracellular solution, as will be shown below, is 27.748 times thicker than<br />
the nucleus whose scaled radius is 1: If we denote the scaled radius of the entire<br />
cell by r, and that of the medium surrounding it by R, then equating the ratio of<br />
their volumes to 168, we have<br />
R3 = 168 (5.1)<br />
3<br />
r<br />
If we then substitute r by 4 in (5.1), we find R to be 22, and therefore the scaled<br />
thickness of the external solution alone is 22 − 4 = 18.<br />
Therefore our computational domain consists of:<br />
• A rectangle which represents the extracellular solution<br />
• A thin outer arc which represents the cell membrane<br />
• A thicker inner arc which represents the cytoplasm. The thousand of tiny<br />
membranes in the cytoplasm are not represented in the domain because they<br />
are handled by homogenisation.<br />
• A thin inner arc which represents the nuclear membrane<br />
• Finally, a thicker central circle which represents the nucleus<br />
All the arcs are concentric with the central circle and only 22.5 o is taken, from<br />
the centre of the central circle, as seen in Figure 5.1.<br />
In the <strong>FEMLAB</strong> draw mode, the scaled domain in Figure 5.1 is drawn. The<br />
representation of a domain in <strong>FEMLAB</strong> is called a geometry [8, p.157].<br />
5.3 Subdomain properties, equations and constants<br />
From the multiphysics menu, a physics is chosen depending on which subdomain,<br />
and appropriate parameters are entered in the subdomain settings dialog box<br />
[8, p.157] to specify the diffusion-reaction equation in that subdomain. The coefficient<br />
form of the PDE mode is used.<br />
16
Figure 5.1. The computational Domain<br />
5.4 Constants and Parameters<br />
It is most convenient to have all the variables and expressions defined in the options<br />
menu [10, p.98] . This is so that if we want to change any coefficients or material<br />
properties, we do not need to go to the subdomain or boundary settings and modify<br />
these for each subdomain. We just need to modify the value by going to the options<br />
menu. In order to know what the constants in my programme stand for, below is a<br />
table of them:<br />
1<br />
5.5 Subdomains and their properties<br />
Subdomain properties of the appropriate physics are presented in Table 5.2. The<br />
boundaries are numbered anti-clockwise round each subdomain. We begin from<br />
boundary 11, which is the left boundary of the extracellular water. The subscripts<br />
indicate the subdomains. Note that for simplicity, the boundary and sub-domain<br />
numbering in this report is not the same as in the programme.<br />
1 The asterices (∗)in Table 5.1 indicate given data. This is the data in the programme that may be<br />
changed by the experimenter. The rest of the entries in the table are computed by the programme.<br />
17
constant meaning and units value<br />
conc Initial concentration in the extracellular solution (M)∗ 1 × 10 −4<br />
cr Fraction of C undergoing chemical reaction 347 × 10 −6<br />
cscale Scaling factor for concentration * 1 × 10 −4<br />
c0 Normalised Initial concentration in the medium 1.0<br />
Dext Diffusion coefficient (D) in the extra-cellular solution � m 2 s −1� ∗ 1.3 × 10 −11<br />
D1 D of the species in the outer membrane � m2s−1� ∗ 1 × 10−12 Effective diffusion coefficient in the cytoplasm � M · m2s−1� 3.9517 × 10−11 D2<br />
D2T Transverse D of the species in the membrane � m 2 s −1� ∗ 1 × 10 −10<br />
D2P Normal D of the species in the membrane � m 2 s −1� ∗ 1 × 10 −12<br />
D3 D of the species in the nuclear membrane � m 2 s −1� ∗ 1 × 10 −12<br />
D4 D of the species in the nucleus � −2s−1 � ∗ 1 × 10−14 D in homogenised cytoplasm if in parallel � m2s−1� 5.926 × 10−11 Dseries<br />
D in homogenised cytoplasm if in series � m2s−1� 2.419 × 10−14 f rac1 Volume fraction of cytoplasm occupied by lipid part * 0.592593<br />
f rac2 Volume fraction of occupied by acquous part 1 − 0.592593<br />
G Concentration of GST (M)∗ 347 × 10−6 Dparallel<br />
Kc Catalytic activity � M−1 · s−1� Kcc<br />
∗<br />
�<br />
−1 Reaction constant of C with GST in cytoplasm = G · Kc s<br />
660003<br />
�<br />
Ku Reaction constant of C with water<br />
2.244<br />
� s−1� ∗ 3.6 × 10−4 M Mass transfer coefficient * 1 × 10−4 n Number of cells whose volume was measured as Vn∗ 1 × 107 N Number of cells in culture * 2 × 107 num s f + sp 3<br />
pump factor determining the rate at which B is pumped out ∗ 0.02<br />
pk Partition coefficient between aqueous and lipid parts ∗ 1 × 10−4 pk2 Partition coefficient between membrane and homogenised cytoplasm 1 × 10 −4<br />
Rcell Radius of cell (m) 1.92 × 10 −4<br />
Rw Scaled radius of part of medium containing cell (m) Computed by code<br />
s Radius of nucleus. Space scaling factor for cell (m)∗ 4.8 × 10 −6<br />
s f Portion of membranes in cytoplasm parallel to diffusion path 2<br />
p f Portion of membranes in cytoplasm normal to diffusion path 1<br />
S Scaling factor of extracellular space in addition to s (m) 55.496<br />
theta Angle of sector of circle representing the cell (radians) π 8<br />
Tws Scaled hickness of part of medium enclosing one cell (m) Computed by code<br />
Tw Thickness of part of medium enclosing one cell (m) Computed by code<br />
Vw<br />
Volume of part medium enclosing one cell � m3� 1 × 10−5 Vn Volume of n cells � m3� ∗ 3 × 10−6 Volume of one cell � m3� Computed by code<br />
V1<br />
Vratio Volume ratio of medium per cell Computed by code<br />
wscale Total scaling factor for extracellular space (m) 1.3874 × 10−4 Table 5.1. Constants used in the programme<br />
18
Physical subdomain Femlab Subdomain Variables boundaries<br />
Water 1 C1, U1 11, 21,31, 41<br />
Cellmembrane 2 C2, U2 32, 52,62, 72<br />
Cytoplasm 3 C3, B3, U3 63, 83,93, 103<br />
Nuclearmembrane 4 C4, U4 94, 114,124, 134<br />
Nucleus 5 C5, A5, U5 125, 145,155<br />
5.6 Initial condition<br />
Table 5.2. Subdomains<br />
In the sub-domamain settings mode, all initial concentrations were left at their default<br />
value, which is 0, except that the scaled initial concentration of C in subdomain<br />
1 was changed to c0.<br />
5.7 Boundary conditions<br />
In the boundary settings dialog box [12] all the exterior boundaries of the domain<br />
were left at their default (insulation). Then choosing the appropriate physics in<br />
<strong>FEMLAB</strong>, the default insulation was again left unchanged for species which do<br />
not diffuse out of their subdomains. These are A and B. Then for C and U, the flux<br />
was set according to (2.10). This is similar to separation through dialysis model<br />
of the <strong>FEMLAB</strong> model library [3, p.213]. While all boundaries are insulated for A<br />
and B, table 3 below sumarises the boundary conditions for C and U.<br />
5.8 Solving<br />
Boundary number Type<br />
subscripts 3, 6, 9, 12 flux<br />
remaining numbers insulation<br />
Table 5.3. Boundary conditions for C and U<br />
The problem was solved with the default solver parameters. It was sufficient to<br />
solve the problem up to 250 seconds.<br />
Solution time was only 1 minute, due to the efficiency of the programme thanks<br />
to homogenisation.<br />
19
5.9 Results<br />
In Figure 5.2, concentrations inside the cytoplasm with time, are compared to the<br />
in vitro results. The concentrations from the in vitro experiments were taken only<br />
at a limited number of instants because it is a very difficult task. The scanty dots<br />
represent concentrations from in vitro experiments, while the graphs are from the<br />
simulations. It is not the actual concentrations which are plotted here, but the percentage,<br />
where the maximum is set to 100 %. Overall, the patterns are similar. The<br />
differences seen might be explained by that the molecular dynamics within the cell<br />
is more complex than we assume in our model.<br />
Figure 5.2. C, B and U in cytoplasm plotted with time<br />
Plots of the concentration of C in the extracellular solution have been compared<br />
to similar plots obtained experimentally by the researchers at K.I. It shows the half<br />
life of C. Please see Figure 5.3.<br />
Laboratory experiments showed that the half life of the substrate C in the solution<br />
was about 63 seconds. The plot in Figure 5.2 obtained from the <strong>FEMLAB</strong> simulation<br />
depicts a half-life of 60 seconds. This is close enough.<br />
20
Figure 5.3. C in solution plotted against time<br />
21
Chapter 6<br />
Conclusion<br />
In the laboratory, careful and tedious measurement techniques are necessary in order<br />
to know the concentration of, for instance, A in the nucleus. Since the model<br />
reproduces those obtained in the laboratory, it can therefore be used as a quick and<br />
easy alternative to determine how much of the carcinogenic product A is present in<br />
the nucleus, which is an indication of the risk of cancer. Here is a plot of the concentration<br />
of A in the nucleus for 500 seconds. One can conclude that the model<br />
can fulfil its aims as indicated in Chapter 1.<br />
The results encourage us to continue developing the model alongside in vitro<br />
experiments.In this model, the chemical reactions occur within the bulk of the aqueous<br />
compartment of the cytoplasm. In future, a similar model is intended to be used<br />
to solve problems where surface reactions are also included. This is the case where<br />
the enzyme which is only present on the surface of the membranes. The present<br />
model will, of course, facilitate the modelling of surface reaction problems, but the<br />
homogenisation of surface reactions are very complex, and a deeper understanding<br />
of the physics and mathematics involved in transport phenomena is needed.<br />
22
Figure 6.1. Concentration of A in the nucleus<br />
23
References<br />
1. K. Sundberg, K. Dreij, A. Seidel and B. Jernstrom.Gluthion Conjugation<br />
and DNA Adduct Formation of Dibenzol [a,1] pyrene and Benzo [a] pyrene<br />
Diol Epoxides in V79 Cells Stably expressing Different Human Glutathione<br />
Transferases. Chemical research in Toxicology ,2002,15, pp. 170-179<br />
2. A. Hartmann, E. M. Golet, S. Gartiser, et al. Primary DNA Damage But Not<br />
Mutagenicity Correlates with Ciprofloxacin Concentrations in German Hospital<br />
Wastewaters Archives of Environmental Contamination and Toxicology,<br />
February 1999, pp. 115 - 119.<br />
3. COMSOL AB, Chemical Engineering Module. 2004, Stockholm, p. 213<br />
4. COMSOL AB, Thin Film Approximation,<br />
http://www.comsol.com/support/knowledgebase/902.php.<br />
5. H. Fredrick, A. Barbero. Steady-state flux and lag time in stratum cornium<br />
lipid pathway using finite element methods. Journal of pharmaceutical Sciences.<br />
Vol 92, NO.11, November 2003, pp. 2196-2207.<br />
6. Lars Erik Persson, Leif Persson, Nols Svanstedt and John Wyller, The Homogenization<br />
Method, an introduction. England, Chartwell Bratt, 1993.<br />
7. Jacob Bear and Yehuda Bachmat, Introduction to Modelling of Transport<br />
Phenomena in Porous Media, The Netherlands, Kluwer Academic , 1990.<br />
8. COMSOL AB, Model Library, <strong>FEMLAB</strong>3. Stockholm, 2004.<br />
9. COMSOL AB, Chemical Engineering Module, Stockholm, 2002, pp. 2-253,<br />
2-254.<br />
10. COMSOL AB, <strong>FEMLAB</strong>3 Users’ Guide. Stockholm, 2004.<br />
24
List of Abbreviations<br />
A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DNA adducts<br />
B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . gluthione conjugates<br />
b.c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . boundary condition<br />
D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . diffusion coefficient<br />
Dar . . . . . . . . . . . . . . . . . . . . . . weighted arithmetic mean diffusion coefficient<br />
D eff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . effective diffusion coefficient<br />
D ha . . . . . . . . . . . . . . . . . . . . . . weighted harmonic mean diffusion coefficient<br />
C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .diol epoxides<br />
GST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . gluthatione transferase<br />
K.I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Karolinska Institutet<br />
Kp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . partition coefficient<br />
REV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . representative elementary volume<br />
U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . tetrols<br />
25