Mathematical Models for Red Blood Cell

Tianrui Xu1* and Charles S. Peskin2

1,2Applied Math Summer Undergraduate Research Experience (AM-SURE), Courant Institute of Mathematical

Sciences, New York University, New York, New York, United States of America*tx392@nyu.edu

Abstract

Red blood cell has a unique biconcave shape because this shape minimizes the bendingenergy of the membrane. In this study, we want to find the simplest discrete model to generatethis shape. We use a triangulated surface to approximate a smoothly continuous surface andre-define curvature on our discrete surface. We formulate an evolution equation involvingbending energy E, volume V , and area A, and use it to produce a shape that is close tobiconcave, but not exactly the same. In the future, we will work on refining our models toreduce errors produced by approximating a smooth surface with discrete method.

1 Introduction

The cell membrane is a very important part of the cell. It not only protects the cell from extremeenvironments and virus invasions, but also connects the cell with the outer world. Nutrients aretransported into the cell through the membrane and waste is transported out. These transportshappen through different chanels on a cell’s membrane and the exocytosis and endocytosis pro-cesses. Armed with such functions, a cell’s membrane helps the cell to communicate with the outerenvironment, although this communication is done without words [1].

Due to the importance of the membrane, we would like to present a mathematical model thatcan generate a cell’s membrane with the right shape. Among all types of cells, red blood cell hasone of the most interesting shapes, which is biconcave. This special shape motivates us to build amodel for the red blood cell to show how it starts from a spherical shape and eventually becomesbiconcave. Although there are many past studies on models of red blood cell, the goal of this projectis to find the simplest model, and to use discrete methods to generate a biconcave cell membranefrom a perfect sphere. This model can be further generalized to other membranes and surfaces thathave unique shapes.

2 Mathematical Modeling

2.1 Initial Condition

To get a biconcave cell, we first start from a sphere and then force the surface to bend. Thediscrete method we use to model a sphere as a triangulated surface, is by dividing the surface intoas many flat triangles as possible. The more triangles we use, the more smooth the surface is [2],

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Figure 1(a) Figure 1(b) Figure 2Figure 1(a)-(b) are examples of using triangulated surface to model a sphere. (a) has a total of 42 points and 80 facesand (b) has a total of 1280 points and 642 faces. The more flat triangles we use, the more smooth the surface is.Figure 2 shows the three vertices with their coordinates Xf1 , Xf2 , and Xf3 , along with the tetrahedron Tf associatedwith the face f .

see figures 1(a) and 1(b). After triangulating the surface, we track the coordinates of every pointswhich are vertices of triangles. To force the membrane to change, we can simply force every points(vertices) to move and then the surface will follow.

In this sphere model, we use 642 points (every vertex of a triangle is a point on the surface)and 1280 faces (every flat triangle is a face). We give a unique index to every point and every face.We store a point by assigning its coordinate to its unique index and we store a face by assigningthe indexes of the three vertices of this triangle to the face’s index. In this work, Xk refers to thecoordinate of the k-th point.

2.2 Model Setup

After initiating the cell membrane from a sphere, we want it to bend to a biconcave shape.According to Canham’s discovery in 1970 [4], the reason a red blood cell is in a biconcave shape isthat the biconcave shape requires the least amount of bending energy. If we allow the membraneof a cell to move freely, it will eventually bend to the biconcave shape to minimize its membrane’sbending energy. Therefore, to obtain the shape of a red blood cell, we first minimize the bendingenergy E, by minimizing dE/dt, where t refers to time. Minimizing dE/dt gives us the directionthat decreases E in the fastest speed.

However, minimizing dE/dt is not enough. If there is no bound on the speed at which E candecrease, we can always find a faster speed by making some points moving faster and making theshape of our membrane changing faster. To achieve a balance between the speed of E and thespeeds of points on our surface, we would like to add a cost when energy decreases faster and thiscost is the speeds of points. Therefore, we would like to minimize the speed of every point, at thesame time with minimizing bending energy E. This means we want to minimize

Q =dE

dt+

1

2

∑k

dXk

dt

dXk

dt,

where the prefactor weight 1/2 is chosen to make dXk/dt and −∂E/∂Xk in equation 1 have thesame weight.

In addition, we want to add area and volume constraints to our equations. To preserve thebasic properties of a cell membrane, we want to keep the surface area A constant. This gives us

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dA/dt = 0. Given a fixed surface area, the shape with the largest volume is a sphere. However,since we start from a sphere and want to keep surface area constant, we have to force volume V todecrease, to ensure possible movements of points in our model. This gives us the second constraintdV/dt = dV0/dt, where V0(t) = V min +(V max−V min)e−at is a given function with negative slopes(V max refers to the initial volume, V min refers to the final volume, and a is a positive constant).

In summary, we would like to minimize Q, subject to dV/dt = dV0/dt and dA/dt = 0. Solvingthis problem, we translate it into the following system of ordinary differential equations (ODEs):

dXk

dt= − ∂E

∂Xk− λ ∂V

∂Xk− µ ∂A

∂Xk, subject to

dV

dt=dV0dt

anddA

dt= 0, (1)

where λ and µ are lagrange multipliers for the constraints.

2.3 Calculation

After setting up the ODE equation, we would like to show how to calculate the bending energyE, the volume V , the surface area A, and the multipliers λ and µ. Bending energy E is defined asE =

∫H2da on a smooth surface, where H and a represent mean curvature and area, respectively.

By first variation of area, we can interpret mean curvature H of a surface as the rate of change ofits area, per unit area, when this surface is moving in the normal direction to itself [5]. Therefore,we would like to apply this definition on our discrete surface to define hf as the mean curvature ofthe face f , see figure 2. Here hf should be the rate of change of face f ’s area, to its area, whenface f is moving in the normal direction to itself. In other words, hf = (daf/dt) · a−1f , and then

E =∫H2da ≈

∑f h

2faf [3]. With regard to calculating hf , we now focus on daf/dt and will

introduce af (the area of the f -th face) in the following paragraph. We first define nf as the unitnormal vector to the f -th face and Nk as the unit normal vector at the k-th point. nf is the vectorperpendicular to the face f and

Nk =

∑i∈F (k) aini∥∥∥∑i∈F (k) aini

∥∥∥ ,where F (k) is the set of indices of faces that meet at the k-th point. In other words, Nk is theweighted average of all normal vectors of faces that meet at the k-th point. Then we are able tocalculate

dafdt

=dafdt

+ 0

=dafdt

nf · nf + afdnfdt· nf (nf is a unit normal vector)

=d

dt(afnf )

=d

dt(Xf1 ×Xf2 +Xf2 ×Xf3 +Xf3 ×Xf1) (see figure 2)

=1

2(Nf1 × (Xf2 −Xf3) +Nf2 × (Xf3 −Xf1) +Nf3 × (Xf1 −Xf2)) , (dXk/dt = Nk by definition of hf )

and finally E.The total volume V =

∑f Vf , where Vf = 1

6Xf1(Xf2×Xf3) is the volume of the tetrahedron Tf ,

see figure 2. The sphere is splited into tetrahedrons, and each tetrahedron Tf is bounded by thef -th triangle face on the membrane’s surface and the center point (the origin). The total surfacearea A is the sum of areas of all surface triangles and can be written as A =

∑f af . af represents

the area of the f -th face (triangle) and we use Heron’s formula af =√s(s− a)(s− b)(s− c), where

s = (a+ b+ c)/2, see figure 2.

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For λ and µ, these are functions of time that ensure both volume and area constraints in equa-tion 1 are followed. To calculate λ and µ, we write the two constraints as

dV

dt=∑k

dV

dXk·dXk

dt

=∑k

dV

dXk·(−∂E

∂Xk− λ

∂V

∂Xk− µ

∂A

∂Xk

)

= −∑k

dV

dXk·∂E

∂Xk− λ

∑k

dV

dXk·∂V

∂Xk− µ

∑k

dV

dXk·∂A

∂Xk

=dV0

dt,

dA

dt=∑k

dA

dXk·dXk

dt

=∑k

dA

dXk·(−∂E

∂Xk− λ

∂V

∂Xk− µ

∂A

∂Xk

)

= −∑k

dA

dXk·∂E

∂Xk− λ

∑k

dA

dXk·∂V

∂Xk− µ

∑k

dA

dXk·∂A

∂Xk

= 0.

This gives the system of equations[∑kdVdXk· ∂V∂Xk

∑kdVdXk· ∂A∂Xk∑

kdAdXk· ∂V∂Xk

∑kdAdXk· ∂A∂Xk

]·[λµ

]=

[−dV0

dt−∑

kdVdXk· ∂E∂Xk

−∑

kdAdXk· ∂E∂Xk

].

2.4 Refinement

Since we are using a discrete method to approach a continuous problem, it becomes importantto keep the surface smooth. Smoothness is important to our model, not only because irregularsurface can produce errors, but also because a cell resists too much local change to its membrane.Therefore, to restrict random movements of points in our model, we want to add more constraintsand penalties to our ODE equation. One method is to restrict the deformation of triangles, i.e.,to avoid too much change on each triangle edge. Therefore, instead of solely minimizing E, weminimize E +α

∑# of edgese=1 l2e , where le represents the length of the i-th edge and α is the penality’s

weight. α is chosen to ensure that both∑

e l2e and E will decrease in a relatively same speed.

Suppose that each triangle is equilateral, then∑

e l2e ≈ 4afne/

√3 = 4af · 3nt/2

√3 ≈ 2

√3A, where

ne is the number of edges and nt is the number of faces. Then α should be 1/(2√

3). By minimizingE +

∑e l

2e/2√

3, we limit the possibility that one edge will get extremely long compared to otheredges, and therefore limit the possibility that triangles will get deformed. Then our ODE equation1 becomes

dXk

dt= − ∂E

∂Xk− 1

2√

3

∂(∑

e l2e

)∂Xk

− λ ∂V∂Xk

− µ ∂A

∂Xk. (2)

We have also tried three other methods to refine our model: (1) forcing points to move only inthe normal direction to the surface; (2) minimizing

∑f a

2f ; and (3) constraining the change of areas

around one specific point. However, the method above, which is to minimize∑

e l2e , works the best

in practice. One possible explanation can be that the squared edge length is related to the elasticpotential energy (Eelastic = 1

2kx2), which strongly resists the change of edge length, and as a result,

helps to keep the shape of every triangle and make the surface smooth.

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Figure 3

Figure 4(a)

Figure 4(b)Figure 3 and 4(a) are viewed from the side. Figure 4(b) is viewed from the top.

2.5 Solving ODEs

We use Euler method to solve equation 2:

Xkn+1 = Xkn + ∆t · dXkn

dt,

where n ∈ N, ∆t is the time step, and Xkn is the coordinate vector of the k-th point at timet = n ·∆t. We apply equation 2 to calculate dXkn/dt. ∆t is chosen to ensure that both volume andarea constraints are followed. If ∆t is too large, it will produce instabilities that break constraints.We use ∆t = 10−6 in our model.

3 Results

Figure 3 shows the result at an early time t = 0.5, and figure 4(a) and 4(b) show the resultat t = 1.8. From figure 4(b), we can see that the surface is perfectly round viewing from the top,which is exactly how red blood cell looks from the same point of view. However, viewing from theside, as shown in figure 4(a), the upper and bottom surfaces haven’t bend inward as expected. Also,the side has formed some acute angles that show a sign of instability. Although the model has notsuccessfully produced a good biconcave, it performs well in following constraints in equations 2. Weplot the total surface area A against time t. The plot shows that A has not changed much in thetime range [0 1.8], so the constraint on area dA/dt = 0 is followed. We also plot the total volumeV against time t. This plot shows that V is changing with V0 curve in the time range [0 1.8], so theconstraint on volume dV/dt = dV0/dt is also followed.

Giving the model more time, we have results at t = 1.9 (figure 5), t = 2.1 (figure 6), and t = 2.3(figure 7). From these figures, we see that the instabilities at the equator in figure 4(a) are getting

Figure 5 Figure 6 Figure 7Figure 5, 6, and 7 are viewed from the side.

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larger. Although these figures show a sign that the upper and bottom surfaces are bending inward,the instabilities at the equator are serious problems that need to be fixed.

4 Conclusion

Using our discrete model, we generate a shape that is close to biconcave, but not exactly thesame. A possible explanation for the instabilities in figure 4(a), 5, 6, and 7 is that our refinement,which ensures the surface is smooth, is not strong enough to resist the deformation of triangles.Another possible explanation is that we fail to choose the right weights (ex. penality weight 1/(2

√3)

in equation 2) or constants (ex. a, the slope of V0 curve) that help to generate the biconcave shape.Therefore, in the future, we will focus on adding more constraints to keep the shapes of surfacetriangles, and changing our weight values and constants. If the future model works on improvingthe smoothness of our discrete model, we will detail our refinement methods and hopefully they canbe useful to other discrete models that are required to keep surface smooth.

5 Acknowledgement

I would like to show my gratitude to my mentor Professor Charles S. Peskin for his great guidanceand encouragement. It is his wisdom that shows me the light through difficulties. I would also liketo present my special thanks to Dr. Pejman Sanaei and Mr. Jason Kaye for their patient assitanceand critiques of the research project. I would also like to thank Professors Miranda Holmes-Cerfonand Aleksandar Donev for leading and supporting the undergraduate research program. Finally, Iwould like to thank NYU Math Department for funding this research.

References

[1] Reece, J. B., Urry, L. A., Cain, M. L. 1., Wasserman, S. A., Minorsky, P. V., Jackson, R., andCampbell, N. A. (2014). Campbell biology. Tenth edition. Boston: Pearson.

[2] Charles S. Peskin. Matlab codes that generate a sphere.

[3] Charles S. Peskin. Notes on defining bending energy on a discrete surface.

[4] Canham, P.B. (1970). The minimum energy of bending as a possible explanation of the bi-concave shape of the human red blood cell. Journal of Theoretical Biology, 26(1), 61-81.doi:10.1016/s0022-5193(70)80032-7.

[5] Sullivan, J.M. (2006). Curvatures of Smooth and Discrete Surfaces. Discrete Differential Geom-etry Oberwolfach Seminars, 175-188. doi:10.1007/978-3-7643-8621-4 9.

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