Approximate Analytical Solution of Chemo Attractant Gradient Sensing Based on Receptor-regulated...

Copyright © 2014 by Modern Scientific Press Company, Florida, USA

International Journal of Modern Mathematical Sciences, 2014, 9(3): 173-197

International Journal of Modern Mathematical Sciences

Journal homepage:www.ModernScientificPress.com/Journals/ijmms.aspx

ISSN:2166-286X

Florida, USA

Article

Approximate Analytical Solution of Chemo Attractant Gradient

Sensing Based on Receptor-regulated Membrane Phospholipid

Signaling Dynamics

K. M. Dharmalingam*, S. Kalaiselvi and V. Ananthaswamy

Department of Mathematics, The Madura College, Madurai-625011, Tamil Nadu, India

* Author to whom correspondence should be addressed: Email: [email protected]

Article history: Received 5 January 2014, Accepted 3 March 2014, Published 10 March 2014.

Abstract: In this paper the analytical solution of steady state non-linear boundary value

problem in phosphoinositide dynamics is discussed. The analytical expression of the

receptor desensitization and the reaction-diffusion process of the phosphoinositide can be

obtained using Homotopy perturbation method (HPM) for various values of the relevant

parameters. In this study, the analytical results have been compared with perturbation

method and which showed that, the present approach has less computational aspect and are

applicable for solving other non-linear boundary value problem.

Keywords: Cell migration; Chemotaxix; Phosophoinositides; Non-linear reaction-diffusion

equations; Homotopy perturbation method; Numerical simulation.

Mathematics Subject Classification (2000): Mathematical modeling and non-linear

differential equation.

1. Introduction

Cellular migration is a coordinated process that resultsfrom the interaction of specific cell

surface receptorswith ligands of the extracellular matrix ~ECM. Usingwell-controlled, stable, ligand

substrates, a number ofligand properties have been shown to affect activation ofcellmotility, including,

for example, ligand surface concentration, strength of ligand–receptor adhesion, degreeof receptor

mailto:[email protected]

Int. J. Modern Math. Sci. 2014, 9(3): 173-197


174

occupancy by the ligand, and ligand affinity[1].The general consensus that, the synthesis of PtdIns is

constitutive and which is the centre of major regulatory mechanisms on the subsequent generationof

the polyphosphoinositides[2]. Actin reorganization is a bidirectional process that orchestratescomplex

cellular events such as, cell migration, neurite extension and bud growth in yeast.The WASP family of

scaffoldingproteins, WASP, N-WASP and WAVE, participate in theseprocesses by relaying signals

from Rho-family GTPases to the actin remodeling machinery [3].Precise regulation of protein-protein

interaction is criticalfor determining the wiring of cellular signaling pathways [4].

Receptors are responsible for transmitting information about theexternal environment to the

cell internum, and therefore theirmechanism of activation and action must be well characterizedbefore

the details of the intracellular networks can be analyzedand modeled mathematically (signaling

network).Two-dimensional molecular gradients in cell membranesare also relevant in signal

transduction. Most signalingpathways involve specific membrane-associated intermediatesthat are

produced or activated through recruitment ofsignaling enzymes to the plasma membrane. Gradients in

thedensity of specific membrane lipids or activated lipidanchoredproteins may form on the nanometer

scale if therates of the reactions that produce them are rapid enough tobe limited by lateral diffusion

and gradients on longer length scales can form whenthe extracellular stimulus is spatially confined or

otherwiseorganized [5].

A fundamental problem of directional sensingis its exquisite sensitivity. Even in the presence

of relativelyshallow chemoattractant gradients, cell projections are extendedprecisely in the region

exposed to the highest chemoattractant concentration. This reflects the existence of a mechanism

foramplifying the external signal.A key problem in directional sensing is the elucidationof the

mechanism underlying the formation of ahighly polarized distribution of actin polymers in responseto

a mild gradient. Indeed, chemoattractant gradients imposed in the extracellular space are often

quitesmall ~1%–2% concentration change over the length of the cell.If a chemo-attractant gradient is

imposed on the cells,GFP–PH migrates within 5–10 s to the leading edge ofthe cell, and persistently

maintains this polarized distribution.This stands in sharp contrast to the results obtainedin response to a

uniform chemo-attractant concentration profile [6].Systems biology modeling of signal transduction

pathwaystraditionally employs ordinary differential equations,deterministic models based on the

assumptions of spatial homogeneity. Cell signaling is an essential, ubiquitous process thatliving

systems use to respond to the environment [7]. Distinguishing modes of Eukaryotic gradient sensing is

developed by Skupsky et.al. [8]. During directional gradient sensing, eukaryotic cells suchas

Dictyostelium and neutrophils exhibit extraordinarysensitivity to external chemical gradients.Shannon

K Hughes-Alford et.al. discussed quantitative analysis of gradient sensing [9]. Noritaka Masaki et.al.

proposed single cell level analysis of PDE expression in Dictyostelium [10].



175

The analytical expression of Ligand bound sensitive receptors, membrane phosphoinositicides,

stored phosphoinositicides and cytosolic inosilol phosphates are presented. These concentrations are

derived using Homotopy perturbation method.

2. Mathematical Formulation of the Problem

To simulate the experiments, it is assumed that before the cell is subjected to a chemoattractant

perturbation ),0( t it is in a homogeneous steady state, ),0,,,( 10 ippr s corresponding to a

uniform chemoattractant concentration, say, l . At ,0t the cell is perturbated by exposing it to a

new chemoattractant concentration profile, .)(l The reduced equations are

2

102

2101010,01

10

10

1

1

r

R

DrKekek

KelKt

r rdta

dl

(1)

2

2

2

2

10

p

R

Dpkcipkpprk

t

p p

pprsf (2)

2

2

2

2

10

sp

pprsfs p

R

Dpkcipkpprk

t

ps (3)

2

2

2

2

10

i

R

Dikcipkpprks

t

i iiirsf (4)

Here, ipspr DDDD and,, denote the lateral diffusivities of ,and,,10 IPPR s and R denotes the cell

radius. The factor s , denoting membrane length per unit cell area, is required since synthesis and

removal rates of P are based on the length of the plasma membrane. Since the cell is circular,

concentrations and fluxes must be equal at 20 and . Thus, we require the periodic boundary

conditions

0,),2(),0(

),,2(),0(

t

txtxtxtx

(5)

where .,,,10 ipprx s The initial conditions for the reduced equations are

iippppp

KelK

KelKrrt ts

dl

dl ,,,1

1;0

10

10

1010

(6)

They reflect the assumptions that:

(1) The total amount of phosphoinositide in the membrane and the endoplasmic reticulum is

conserved, so that the average phosphoinositide concentration, denoted tp is constant.

(2) Information concerning the chemoattractant concentration profile imposed at 0t , namely,

l , is transmitted to the receptors instantaneously. Thus, 10r changes during this rapid

process, but iandpp s ,, remain unchanged.



176

By introducing the dimensionless variables

10

10

10r

r ,

tp

p ,

t

ss

p

p , ι

tsp

i ,

1416.3*2

,

tr spk

t

1

(7)

and dimensionless parameters

tr spk

k01

01 ,

10

,

,r

e ta

ta , lK

l ,

10K

ed

d , tr

rr

spk

CD 2

(8)

tr

tf

fspk

prk 2

10

, tr

tp

pspk

pc ,

tr

p

pspk

k ,

tr

p

pspk

CD 2

(9)

tr

sp

pspk

CD

s

2

,

tr

tii

spk

spc ,

tr

ii

spk

k ,

tr

ii

spk

CD 2

, lK

l

(10)

tp

p ,

t

ss

p

p , ι-

tsp

i

(11)

Where C denotes the circumference of the cell. Thus we get the dimensionless equations of (1)-(6) are

as follows:

2

1

2

111 )1(

XXk

Xr (12)

2

2

2

243423

2

2122

XXkkXXXXXk

Xp

(13)

3X 2

3

2

243423

2

212

XXkkXXXXXk

sp

(14)

2

4

2

465423

2

2124

XXkkXXXXXk

Xr

(15)

with initial conditions

34232211 and1,,,0 uXuXuXuX

(16)

And the periodic boundary conditions

,0,),1(),0(

),,1(),0(

xxxx (17)

where

.,,,10 ipprx s (18)

Here

,,, 32101 sXXX 4X = ι (19)

iippf

d

takkkkkk

65432

,01

1 ,,,,,/11

(20)



177

321 ,,)(/11

/11uuu

d

d

ι- (21)

By considering 02

4

2

2

3

2

2

2

2

2

1

2

XXXX (22)

Then the equations (12)-(15) becomes

)1( 111 Xk

X

(23)

243423

2

2122 XkkXXXXXk

X

(24)

3X 243423

2

212 XkkXXXXXk

(25)

465423

2

2124 XkkXXXXXk

X

(26)

with the initial conditions

34232211 and1,,,0 uXuXuXuX

(27)

3. Solution of the Problem Using Homotopy Perturbation Method

Linear and non-linear phenomena are of fundamental importance in various fields of science

and engineering. Most models of real – life problems are still very difficult to solve. Therefore,

approximate analytical solutions such as Homotopy perturbation method (HPM) [11-24] were

introduced. This method is the most effective and convenient ones for both linear and non-linear

equations. Perturbation method is based on assuming a small parameter. The majority of non-linear

problems, especially those having strong non-linearity, have no small parameters at all and the

approximate solutions obtained by the perturbation methods, in most cases, are valid only for small

values of the small parameter. Generally, the perturbation solutions are uniformly valid as long as a

scientific system parameter is small. However, we cannot rely fully on the approximations, because

there is no criterion on which the small parameter should exists. Thus, it is essential to check the

validity of the approximations numerically and/or experimentally. To overcome these difficulties,

HPM have been proposed recently.

Recently, many authors have applied the Homotopy perturbation method (HPM) to solve the

non-linear problem in physics and engineering sciences [11-14]. This method is also used to solve

some of the non-linear problem in physical sciences [15-17]. This method is a combination of

Homotopy in topology and classic perturbation techniques. Ji-Huan He used to solve the Lighthill

equation [15], the Duffing equation [16] and the Blasius equation [17-18]. The HPM is unique in its



178

applicability, accuracy and efficiency. The HPM uses the imbedding parameter p as a small parameter,

and only a few iterations are needed to search for an asymptotic solution. Using this method [19-24],

we can obtain solution of the eqns. (23) - (27) are as follows:

1)1()( 11 eutX (28)

1

)(

4

3

21232

6

)(

6

43

4

3

2

41

)2(2

4

32122

14

2

4

2

3122

4

22

4

3222

4

6

5

3232

6

52

644

3

6

5

3

6

2

4

53

3

4

2

3221

4

3

4

322

6464

41

1

4

4

6

44

)()1()1(2)()(

)(

)()1()1(

)(

)1()1(

)()1()()1(2

)(

)()1(

)(

k

ek

kuuukk

k

ek

ku

k

ku

kk

ek

kuuuk

kkk

ekuuk

k

ek

kuuk

k

k

kuukk

k

kve

kkk

ekk

ku

kk

kk

k

kuked

k

ke

k

kutX

kkkk

kk

k

k

k

k

kk

(29)

1

)(

4

3

21232

6

)(

6

43

4

3

2

41

)2(2

4

3

2122

14

2

4

2

3122

4

22

4

3

222

4

6

5

3232

6

52

644

3

6

5

3

6

2

4

53

3

4

2

322

1

4

3

4

3

2

3

6464

41

1

4

4

6

44

)()1()1(2)()(

)(

)()1()1(

)(

)1()1(

)()1()()1(2

)(

)()1(

1)(

k

ek

kuuukk

k

ek

ku

k

ku

kk

ek

kuuuk

kkk

ekuuk

k

ek

kuuk

k

k

kuukk

k

kve

kkk

ekk

ku

kk

kk

k

kuked

k

ke

k

ku

tX

kkkk

kk

k

k

k

k

kk

(30)



179

)(

)()1()1(2

)2(

)()1)(1(

)(

)(

)1()1(

)2(

)()1(

)()()()1(2)(

)1()(

4164

)(

4

3

21232

416

)2(2

4

3

2122

4

3

6

5

3

16

2

4

2

3122

46

22

4

3

222

4

)(

6

5

3

4

3

2

4

6

5

3232

6

5

4

3

2

46

2

64

53

6

2

4

2

322

2

6

5

6

5

34

4114

6

1

4

66

4

66

kkkk

ek

kuuukk

kkk

ek

kuuuk

k

etkk

ku

kkk

ekuuk

kk

ek

kuuk

k

ek

ku

k

ku

k

k

kuukk

k

kk

ku

kk

e

kk

kk

kk

kuked

k

ke

k

kutX

kkkk

k

k

k

kk

k

kk

(31)

where 1d and 2d are defined in the Appendix B.

Fig. 1: Illustration of experiment for identifying the firstpolarized component in the signal transduction

pathway.

4. Numerical Simulation

The non-linear differential eqns. (23) - (27) are also solved by numerical methods using Scilab

software. Its numerical solution is compared with Homotopy perturbation method in Figs (3) – (6) and

it gives satisfactory result for all small values of the dimensionless parameters. The Scilab program is

also given in Appendix (C).



180

Fig. 2: The phosphoinositide cycle (a) and the model scheme (b). In the model scheme, R10 denotes

active receptors; P, I, and Ps denote the pools of membrane phosphoinositicides, cytosolic inosilol and

its phosphates, and phosphoinositicides in the endoplasmic reticulum, respectively.

5. Results and Discussions

Eqns.(28-31) are the new approximate analytical expressions for the ligand –bound sensitive

receptors, membrane phosphoinositicides, stored phosphoinositicides and cytosolic inosilol

phosphates. Figs 3,4 shows the evolution of the solutions when the chemoattractant concentration is

increased from 1000to5.0 . The dimensionless concentration of ligand-bound sensitive



181

receptors 1X and dimensionless concentration of membrane phosphoinositicides 2X versus the

dimensionless time are indicated in the Fig. 3.

Fig. 3: Dimensionless concentrations of ligand-bound sensitive receptors 1X and the membrane

phosphoinositicides 2X versus the dimensionless time . The curve is plotted for the various values of

dimensionless ligand concentration , when 200,1,5.0 1000.and

The observation from the Fig. 3, it is evident that when the dimensionless ligand concentrations

increases, the corresponding concentrations 1X and 2X increases. Whereas, the dimensionless

concentration stored phosphoinositicides 3X and the dimensionless inositol phosphates 4X versus the

dimensionless time were represented in Fig. 4.



182

Fig. 4: Dimensionless concentration stored phosphoinositicides 3X and the cytosolic inosilol

phosphates 4X versus the dimensionless time . The curve is plotted for the various values of

dimensionless ligand concentration , when 1000.and200,1,5.0

From the Fig. 4, it is inferred that, when the dimensionless ligand concentration increases,

there was a correspond decrease in the concentration of 3X, and the concentrations of 4X increases

respectively. The ligand bound receptors concentration and membrane phosphoinositicides

concentration is plotted in Fig.5 is a particular case when 2001 and .



183

Fig. 5: Dimensionless concentrations of ligand-bound sensitive receptors 1X and the membrane

phosphoinositicides 2X versus the dimensionless time . The curve is plotted for the various values of

dimensionless ligand concentration , when 200.and1

In Fig.6 the concentrations of stored phosphoinositicides and concentrations of cytosolic

inosilol phosphates are varying according to the Ligand concentration. From Fig.7and Fig.8 it is



184

observed that the concentration of membrane phosphoinositicides and cytosolic inosilol phosphates are

coinciding with respect to the ligand concentration.

Fig. 6: Dimensionless concentration stored phosphoinositicides 3X and the cytosolic inosilol


dimensionless ligand concentration , when 200.and1



185

Fig. 7: Dimensionless concentrations of ligand-bound sensitive receptors 1X , the membrane

phosphoinositicides 2X , the stored phosphoinositicides 3X and the cytosolicinosilol


dimensionless ligand concentration , when 1.and5.0



186

Fig. 8: Dimensionless concentrations of ligand-bound sensitive receptors 1X , the

phosphoinositides 2X , the stored phosphoinositicides 3X and the inositol phosphates 4X versus the

dimensionless time . The curve is plotted for the various values of dimensionless ligand

concentration , when 1000.and200



187

6. Conclusions

It has been observed and concluded that, a reaction-diffusion model based on the kinetics of the

phosphoinositicides cycle captures and explains many of the phosphoinositicides dynamics associated

with directional sensing in cell migration. The analytical expressions of the dimensionless

concentrations of ligand-bound sensitive receptors, membrane phosphoinositicides, stored

phosphoinositicides in endoplasmic reticulum and inosilol phosphates for all the small values of the

corresponding dimensionless parameters. The analytical results are compared to the numerical

simulations. The HPM is an extremely simple compared to other method and it is also a promising

method to solve other non-linear equations. This method can be easily extended to find the solution of

all other non-linear equations.

Acknowledgement

The authors are thankful to Shri. S. Natanagopal, Secretary, Madura College Board, Madurai

and Dr. R. Murali, The Principal for their constant encouragement.

References

[1] JanesS. Tjia and Prabhas V. Moghe, Cell Migration on Cell-Internalizable Ligand Microdepots:

A Phenomenological Model, Annals of Biomedical Engineering, 30(2002):851–866.

[2] Samer J. Nuwayhid, Martha Vega, Paul D. Walden, Marie E. Monaco1,Regulation of de novo

phosphatidylinositol synthesisThe WRP component of the WAVE-1complex attenuates Rac-

mediated signaling, Journal of lipid research, 47 (2006): 1449-1456.

[3] Scott H. Soderling, Kathleen L. Binns, Gary A. Wayman, Stephen M. Davee, Siew Hwa Ong,

Tony Pawson, John D. ScottThe WRP component of the WAVE-1complex attenuates Rac-

mediated signaling, Nature cell Biology,4 (2003): 970-975.

[4] Nathan A. Sallee, Brian J. Yeh, and Wendell A. Lim, Engineering Modular ProteinInteraction

Switches bySequence Overlap, Biophysical Journal, 86 (2004): 589–598.

[5] Jason M. Haugh and Ian C. Schneider, Spatial Analysis of 39 Phosphoinositide Signaling in

Living Fibroblasts: I. Uniform Stimulation Model and Bounds on Dimensionless Groups,Bio

Physical Journal, 86 (2004): 589-598.

[6] Atulnarang, K. K. Subramanian,and D. A. Lauffenburger, A Mathematical Model for

Chemoattractant Gradient Sensing Based on Receptor-Regulated Membrane

PhospholipidSignaling Dynamics, Annals of Biomedical Engineering, 29 (2001): 677-691.



188

[7] Krishnan Radhakrishnan, A´da´mHala´sz, Dion Vlachos, Jeremy S Edwards, Quantitative

understanding of cell signaling: the importance ofmembrane organization, Current Opinion in

Biotechnology, 21(2010):677–682.

[8] R.Skupsky, W. Losert, R.J. Nossal, Distinguishing Modes of Eukaryotic gradient sensing,

Biophysical Journal, 89(2005): 2806-2823.

[9] S. K Hughes-Alford, D. A Lauffenburger, Quantitative analysis of gradient sensing: towards

building predictive models of chemotaxis in cancer, current opinion in cell Biology, 24(2012):

284-291.

[10] N. Masaki, K. Fujimoto, M. Honda-Kitahara, E. Hada and S. Sawai, Robustness of Self-

Organizing Chemoattractant Field Arising from Precise Pulse Induction of its Breakdown

Enzyme: A Single-Cell Level Analysis of PDE Expression in Dictyostelium, Biophysical

Journal, 104(2013): 1191-1202.

[11] Q.K. Ghori, M. Ahmed, and A. M. Siddiqui, Application of Homotopy perturbation method to

squeezing flow of a Newtonian fluid, Int. J. Nonlinear Sci. Numer. Simulat, 8(2007): 179-184.

[12] T. Ozis, and A. Yildirim, A Comparative study of He’s Homotopy perturbation method for

determining frequency-amplitude relation of a nonlinear oscillator with discontinuities,Int. J.

Nonlinear Sci. Numer. Simulat, 8 (2007): 243-248.

[13] S. J. Li, and Y. X. Liu, An Improved approach to nonlinear dynamical systemidentification

using PID neural networks, Int. J. Nonlinear Sci. Numer. Simulat, 7 (2006): 177-182.

[14] M. M. Mousa, S.F. Ragab, and Z. Nturforsch, Application of the Homotopy perturbationmethod

to linear and nonlinear Schrödinger equations,. Zeitschrift für Naturforschung, 63(2008): 140-

144.

[15] J.H. He, Homotopy perturbation technique, Comp Meth. Appl. Mech. Eng, 178(1999): 257-262.

[16] J. H. He, Homotopy perturbation method: a new nonlinear analytical technique, Appl.Math.

Comput, 135(2003): 73-79.

[17] J. H. He., A simple perturbation approach to Blasius equation, Appl. Math. Comput, 140(2003):

217-222.

[18] P.D. Ariel, Alternative approaches to construction of Homotopy perturbation Algorithms, Non-

linear. Sci. Letts. A., 1(2010): 43-52.

[19] V. Ananthaswamy, SP. Ganesan and L. Rajendran, Approximated analytical solution ofnon-

linear boundary value problem in steady state flow of a liquid film: Homotopy perturbation

method, International Journal of Applied Sciences and Engineering Research(IJASER), 5

(2)(2013):569- 577.



189

[20] A. Meena and L Rajendran, Mathematical modeling of amperometric and potentiometric

biosensors and system of non-linear equations –Homotopy perturbation approach, J. Electroanal

Chem, 644(2010): 50-59.

[21] S. Anitha, A. Subbiah, S. Subramaniam and L. Rajendran, Analytical solution of amperometric

enzymatic reactions based on Homotopy perturbation method, Electrochimica Acta, 56(2011):

3345-3352.

[22] V. Ananthaswamy and L. Rajendran, Analytical solution of two-point non-linear boundary value

problems in a porous catalyst particles, International Journal of Mathematical Archive, 3

(3)(2012): 810-821.

[23] V. Ananthaswamy and L. Rajendran, Analytical solutions of some two-point non-linear elliptic

boundary value problems, Applied Mathematics, 3(2012): 1044-1058.

[24] V. Ananthaswamy and L. Rajendran, Analytical solution of non-isothermal diffusion-reaction

processes and effectiveness factors, ISRN Physical Chemistry, Article ID 487240, 2012: 1-14.



190

Appendix A

Basic concept of Homotopy perturbation method [11-24]

To explain this method, let us consider the following function:

r ,0)()( rfuDo (A. 1)

with the boundary conditions of

r ,0) ,(

n

uuBo (A. 2)

where oD is a general differential operator, oB is a boundary operator, )(rf is a known analytical

function and is the boundary of the domain . In general, the operator oD can be divided into a

linear part L and a non-linear part N . Eq. (A.1) can therefore be written as

0)()()( rfuNuL (A. 3)

By the Homotopy technique, we construct a Homotopy ]1,0[:),( prv that satisfies

.0)]()([)]()()[1(),( 0 rfvDpuLvLppvH o (A. 4)

.0)]()([)()()(),( 00 rfvNpupLuLvLpvH (A. 5)

wherep [0, 1] is an embedding parameter, and 0u is an initial approximation of Eq. (A. 1) that

satisfies the boundary conditions. From Eq. (A.4) and Eq. (A.5), we have

0)()()0,( 0 uLvLvH (A. 6)

0)()()1,( rfvDvH o (A. 7)

When p=0, Eq. (A.4) and Eq. (A.5) become linear equations. When p =1, they become non-linear

equations. The process of changing p from zero to unity is that of 0)()( 0 uLvL to 0)()( rfvDo .

We first use the embedding parameter p as a “small parameter” and assume that the solutions of Eq.

(A.4) and Eq. (A.5) can be written as a power series in p :

...2

2

10 vppvvv (A. 8)

Setting 1p results in the approximate solution of Eq. (A.1):

...lim 2101

vvvvup

(A. 9)

This is the basic idea of the HPM.



191

Appendix B

Solution of the boundary value problem eqns. (23) - (27) using Homotopy perturbation method.

In this Appendix, we indicate how eqns. (28) - (31) are derived in this paper. Solving eqns. (23) and

(27) we get the solution is

1)1( 11 euX (B.1)

To find the solution of the eqns. (24) – (27), we construct the Homotopy as follows:

0)1( 3423

2

212242

3242

kXXXXXkXk

dt

dxpkXk

dt

dXp (B.2)

0)1( 3423

2

212243

3243

kXXXXXkXk

dt

dXpkXk

dt

dXp (B.3)

0)1( 5423

2

212464

5464

kXXXXXkXk

dt

dXpkXk

dt

dXp (B.4)

The analytical solutions of (B.2), (B.3) and (B.4) is

..........210 22

2

21202 XpXpXX (B.5)

.........210 32

2

31303 XppXXX (B.6)

.........210 42

2

41404 XppXXX (B.7)

Substituting the eqns. (B.5), (B.6) and (B.7) in (B.2), (B.3) and (B.4) respectively we get

0

.....).....)((

.....)(.....)(

.....)(.....)(

.....)(.....)(

)1(

342

2

414022

2

2120

32

2

3130

2

22

2

212012

22

2

21204

22

2

2120

322

2

21204

22

2

2120

210210

210210

210

210

210

210

kXpXpXXpXpX

XpXpXXpXpXXk

XpXpXkdt

XpXpXd

p

kXpXpXkdt

XpXpXdp

(B.8)

0

.....).....)((

.....)(.....)(

.....)(.....)(

.....)(.....)(

)1(

342

2

414022

2

2120

32

2

3130

2

22

2

212012

22

2

21204

32

2

3130

322

2

21204

32

2

3130

210210

210210

210

210

210

210

kXpXpXXpXpX

XpXpXXpXpXXk

XpXpXkdt

XpXpXd

p

kxpXpXkdt

XpXpXdp

(B.9)



192

0

.....).....)((

.....)(.....)(

.....)(.....)(

.....)(.....)(

)1(

542

2

414022

2

2120

32

2

3130

2

22

2

212012

42

2

41406

42

2

4140

542

2

41406

42

2

4140

210210

210210

210

210

210

210

kXpXpXXpXpX

XpXpXXpXpXXk

XpXpXkdt

XpXpXd

p

kXpXpXkdt

XpXpXdp

(B.10)

Comparing the coefficients of like powers of p in (B.8), (B.9) and (B.10) we get

0: 3204

200

0 kXk

dt

Xdp (B.11)

0: 3

300 kdt

Xdp (B.12)

0: 5406

400

0 kXk

dt

dXp (B.13)

402030

2

0212214211 : XXXXXkXk

dt

dXp (B.14)

402030

2

0212214

311 : XXXxXkXkdt

dXp (B.15)

402030

2

0212216411 : xxxxxkxk

dt

dxp (B.16)

The initial approximations are as follows

340230220110 )0(and1)0(,)0(,)0( uXuXuXuX (B.17)

,0)0()0()0()0( 4321 iiii XXXX ......3,2,1i (B.18)

Solving the eqns. (B.11) - (B.16) and using the initial conditions (B.17) and (B.18) we obtain the

following results:



193

(B.19)4

3

4

3220

4

k

ke

k

kuX

k

1

)(

4

321232

6

)(

6

43

4

32

41

)2(2

4

32122

644

3

6

53

1424

23122

4

22

4

3222

4

6

53232

6

52

624

53

34

2322

121

64

6441

6

1

4

44

)()1()1(2

)()(

)(

)()1()1(

)(

)(

)(

)1()1()()1(

)()1(2)1(

k

ek

kuuukk

k

ek

ku

k

ku

kk

ek

kuuuk

kkk

ekk

ku

kkk

ekuuk

k

ek

kuuk

k

k

kuukk

k

kvet

kk

kk

k

kukedX

kk

kkkk

k

k

k

kk

(B.20)

Where

1

4

321232

6

6

43

4

32

41

2

4

32122

644

3

6

53

1424

23122

4

2

4

3222

34

2322

624

531

)()1()1(2

)()(

)(

)()1()1(

)(

)(

)(

)1()1()()1(

)1(

k

k

kuuukk

k

k

ku

k

ku

kk

k

kuuuk

kkk

kk

ku

kkk

kuuk

k

k

kuuk

k

kuk

kk

kkd

(B.21)

Since ,032 d

dX

d

dX we get

23 1 XX

(B.22)



194

(B.23)6

5

6

5340

6

k

ke

k

kuX

k

(B.24)

)()(

)(

)()1()1(2

)2(

)()1)(1(

)(

)(

)1()1(

)2(

)()1(

)()1(2)()1(

4

)(

6

53

4

32

4164

)(

4

321232

416

)2(2

4

32122

4

3

6

53

1624

23122

46

22

4

3222

4

6

53232

6

5

4

32

46264

53

624

2322

241

6641

14

6

1

4

46

k

ek

ku

k

ku

kkkk

ek

kuuukk

kkk

ek

kuuuk

k

ekk

ku

kkk

ekuuk

kk

ek

kuuk

k

k

kuukk

k

kk

ku

kk

e

kk

kk

kk

kukedX

kkkk

kk

k

k

k

kk

where

4

6

53

4

32

4164

4

321232

416

2

4

32122

4

3

6

53

1624

23122

46

2

4

3222

4

6

53232

6

5

4

32

46624

2322

264

532

)()(

)(

)()1()1(2

)2(

)()1)(1(

)(

)(

)1()1(

)2(

)()1(

)()1(2)(1)1(

6

k

k

ku

k

ku

kkkk

k

kuuukk

kkk

k

kuuuk

k

etkk

ku

kkk

kuuk

kk

k

kuuk

k

k

kuukk

k

kk

ku

kkkk

kuk

kk

kkd

tk

(B.25)

According to the HPM, we can conclude that

)(lim10 21202

12 XXtXX

p

(B.26)



195

1 23 XX (B.27)

)(lim10 41404

14 XXtXX

p

(B.28)

After putting the eqns. (B.19) and (B.20) into an eqn. (B.26) and (B.23) and (B.24) into an eqn.

(B.28), we obtain the solutions in the text eqns. (29) and (31).

Appendix C

Scilab program to find the solutions of the Equations (23)-(27)

function

options= odeset('RelTol',1e-6,'Stats','on');

%initial conditions

x0 = [10.08174387; .1; .9; .1];

tspan = [0,200];

tic

[t,x] = ode45(@TestFunction,tspan,x0,options);

toc

figure

hold on

%plot(t, x(:,1))

%plot(t, x(:,2))

plot(t, x(:,3))

%plot(t, x(:,4))

legend('x','y','z')

ylabel('x')

xlabel('t')

return

function [dx_dt]= TestFunction(t,x)

k1=.09523809524;k2=1.1;k3=.01;k4=.1;k5=.01;k6=.11;

dx_dt(1)=k1*(1-x(1));

dx_dt(2)=k2*x(1)*x(2)^2*x(3)-x(2)*x(4)+k3-k4*x(2);

dx_dt(3)=-(k2*x(1)*x(2)^2*x(3)-x(2)*x(4)+k3-k4*x(2));

dx_dt(4)=k2*x(1)*x(2)^2*x(3)-x(2)*x(4)+k5-k6*x(4);

dx_dt = dx_dt';

return



196

Appendix D

Nomenclature

Symbols Meaning

ija Rate constants for association of aE or dE with ijR

la Rate constants for association of ijR with L

ic Basal synthesis rate of synthesis of inosital phosphates

c Circumference of the cell

iD Lateral diffusivity of inosital phosphates

pD Lateral diffusivity of phoshoinositides in plasma membrane

spD Lateral diffusivity of phoshoinositides in endoplasmic

reticulum

rD Lateral diffusivity of receptors

tae , Total concentration of activating (dephosphorylating ) enzyme

de Concentration of deactivating (phosphorylating ) enzyme

i Concentration of inositol phosphates

fk Rate constant for receptor-mediated phoshoinositide formation

ik Rate constant for basal degradation of inositolphosphates

pk Rate constant for basal degradation of phosphoinositides

rk Rate constant for removal of phosphoinositides

ijk Rate constants for sensitization/ desensitization

ijK Dissociation constants for binding of aE or dE to ijR

lK Dissociation constant for ligand binding

l Concentration of ligand

p Concentration of membrane phosphoinositides

sp Concentration of phosphoinositides on endoplasmic reticulum

lp Average concentration of phosphoinositides in the cell

ijr Concentration of receptors with i ligand-bound and j

phosphorylated sites

*ij

r Concentration of complexes between aE or dE and ijR

fpr , Rate of formation of membrane phosphoinositides

rpr , Rate of removal of membrane phosphoinositides

dpr , Degradation rate of membrane phosphoinositides

dir , Degradation rate of inositol phosphates

R Radius of the cell

s Membrane length per unit cell area



197

p Dimensionless rate of basal synthesis of phosphoinositides

i Dimensionless rate of basal synthesis of inositol phosphates

i Dimensionless angular diffusivity of inositol phosphates

p Dimensionless angular diffusivity of phosphoinositides in

plasma membrane

sp Dimensionless angular diffusivity of phosphoinositides in

endoplasmic reticulum

r Dimensionless angular diffusivity of receptors

Ι Dimensionless concentration of inositol phosphates

ta, Dimensionless concentration of activating enzyme

d Dimensionless concentration of deactivating enzyme

f Dimensionless rate constant for formation of

phosphoinositides

i Dimensionless rate constant for degradation of inositol

phosphates

p Dimensionless rate constant for degradation of

phosphoinositides

Dimensionless ligand concentration

Dimensionless concentration of phosphoinositides

s Dimensionless concentration stored phosphoinositicides

10 Dimensionless concentration of ligand-bound sensitive

receptors

Relative gradient of chemoattractant

Dimensionless time

Angular coordinate scaled between 0 and 1

Approximate Analytical Solution of Chemo Attractant Gradient Sensing Based on Receptor-regulated...

Documents

Transcript of Approximate Analytical Solution of Chemo Attractant Gradient Sensing Based on Receptor-regulated...