A Mathematical Model of Cell Reprogramming due to Intermediate Differential Regulator's Regulations
AA Mathematical Model of Cell Reprogrammingdue to Intermediate Differential Regulator’sRegulations
Arnab BaruaDepartment of Physics, IIT BombayEmail:[email protected] 19, 2018
Abstract
In this paper I have given a mathematical model of Cell reprogrammingfrom a different contexts. Here I considered there is a delay in differentialregulator rate equations due to intermediate regulator’s regulations. Atfirst I gave some basic mathematical models by Ferell Jr.[2] of reprogram-ming and after that I gave mathematical model of cell reprogramming byMithun Mitra[4]. In the last section I contributed a mathematical modelof cell reprogramming from intermediate steps regulations and tried tofind the critical point of pluripotent cell .
Basic unit of biological organisms are cells. In mammals sperm fertilize the eggand make an embryonic stem cell, which is a single cell. A beautiful thing inbiological system is that all the biological organisms are made from the singlecell. We can find 200 types of cells in our body and their jobs are different.They have same DNA sequence. This phenomena is known as Cell Differentia-tion.Where a cell type changes another cell type. Cell signals play a major rolein this context externally and internally. So, the question is that containingsame kind of information how will you get different kind of cell types in yourbody?Can we understand this biological process from dynamical systems pointof view?Another beautiful phenomena is known as Cell reprogramming where the in-formation contained in cell is removed and due to external stimulus it movesinto a different cell type. For this John Gurdon and Shinya Yamanaka got theNoble Prize in 2012. In this paper I studied the Cell differentiation and CellReprogramming phenomena from dynamical systems point of view.1 a r X i v : . [ q - b i o . CB ] J un An Intuitive Picture and Biology of Cell Dif-ferentiation
At first I will give an intuitive picture. So let you and your friend went to alibrary and borrowed some books. Say your favorite subject is Physics and yourfriend’s favorite subject is History. When both of you read this book , your mindwill definitely change.But all the books are in the library. So in biology languageCell differentiation phenomena is that in the same DNA sequence some genesare off and some are on (say 1011 is for eye cell and 1100 for Lip cell,1 meanson 0 means off). It happens due cues which can be act internally or externally.DNA Methylation, Histone Acetylation, mRNA these play an important role inCell Differentiation.
In 1957 C.H.Waddington[1] gave an intuitive picture where he considered Cells(asa balls)rolling down in a potential surface(as Hills and valleys). Where each val-ley is considered as a cell fate. Ferell Jr.[2] proposed some mathematical modelswhich is quite similar with Waddington’s Epigenetic Landscape. two modelsare 1)Cell fate induction model 2) Lateral inhibition Model.
A cell can induce another cell to be differentiated and adopt a phenotype(or a group of cells can induce another group of cells to be differentiated andadopt a phenotype) by chemical signals. In this model he considered a singledifferential regulator of concentration x (Example- Transcription factors) whichhas a constant basal rate α , a maximum positive hill function for feedback rateof synthesis x is β and a degradation rate γ .The rate of concentration x is (Considering a high coefficient n = 5) dxdt = α + β (cid:32) x K + x (cid:33) − γx where K is the concentration of x when the feedback is half of the maximum.The potential function is φ and dφdx is for the speed at which it’s come tosteady state.So, φ = − (cid:90) (cid:16) α + β ( x K + x ) − γx (cid:17) dx After Plotting this potential function φ vs. x, there is two stable points atx=0 and x= 1.7 in Fig 1. and an unstable point at x=1. If you increase thevalue of α and β , you will find that there is only one steady state(Fig 2.). So,it’s a Saddle-node bifurcation.Also, it’s different from Waddington’s EpigeneticLandscape.Basic difference with Waddington’s Landscape is that in Cell Fate Model thevalley is created or destroyed for the value of α and β . But in Waddington’sLandscape the valley’s are permanent but Cell will take the decision internallyor externally at the critical point.Potential function( φ )(y axis) vs. x (x axis) plot.2ic1( α =0, β =1, γ =0.5,K=1)Pic2( α =1, β =1, γ =0.5,K=1) In this model he considered that two daughter cells m and n made from themother cell x.The daughter cells m and n will mutually inhibit each other. Theinput signal of this model is interaction (I). After a critical value of interac-tion(I)=0.5 either m will win or n will win. He got Waddington’s Landscape ofPitchfork Bifurcation. The mathematical model is dmdt = α (cid:32) K K + ( In ) (cid:33) − βmdndt = α (cid:32) K K + ( Im ) (cid:33) − βn where α is the coefficient of inhibition, β is the coefficient of degradation and Kis the concentration of x when the feedback is half of the maximum. It’s alsodepends on initial conditions of two daughter cells. Cell reprogramming is a phenomena where the epigenetic memory of a cellfate is removed by some stimulus and we can get another cell fate. Such anexperiment was done by Nagy and Nagy[3]. They did this experiment and founddifferentiated fibroblast cells that were derived from induced pluripotency andalso got four Yamanaka Factors under the control of Doxycycline drug.3 .1 Mathematical model for Cell Reprogramming
Mithun Mitra and their group[4] proposed a mathematical model of cellularreprogramming on the basis of Ferell’s Cell Fate Model. They said that ifwe apply the Doxycycline for a certain time then we can get Pluripotent Cellfrom Somatic Cell. Also they considered the delay time for multiple chemicalreactions,Cell shape and physical origins. They didn’t take the delay term fordegradation part. The mathematical model is dxdt = α [ d − t ] + β (cid:32) ( x − p ) K + ( x − p ) (cid:33) − γx Where α is the concentration of Doxycycline drug, β is the coefficient of inhi-bition and γ is degradation coefficient. p is the time of delay and d is the timeup to which the Doxycycline drug is applied to the cell. First term here theyconsidered the Heaviside step function for constant supply of drug upto certaintime.Pic3( α =0.5, β =1, γ =0.5,K=1,p=0,d=2),Initial condition(x[0]=0) they defined it’sSomatic cell at x=0.Because in Ferell’s Cell Fate Model there are two stablestates at x=0 and x= 1.7Pic3( α =0.5, β =1, γ =0.5,K=1,p=0,d=3),Initial condition(x[0]=0) It’s the Pluripo-tent Cell. But beautiful thing is that I find some meta stable states at x=4 for4articular α , β and γ shown in these pictures( α =1, β =1, γ =0.5,K=1,p=0,d=250),Initial condition(x[0]=10)( α =1, β =1, γ =0.5,K=1,p=0,d=250),Initial condition(x[0]=0.5) I proposed a mathematical model where I considered intermediate regulation inX(where some differential regulator of x will regulate differential regulator n anddifferential regulator m will regulate differential regulator n .And the differentialregulator n will regulate differential regulator x). My mathematical model is dxdt = α [ d − t ] + β (cid:32) m K + m (cid:33) − γxdydt = β (cid:32) x K + x (cid:33) − γydzdt = β (cid:32) y K + y (cid:33) − γzdmdt = β (cid:32) z K + z (cid:33) − γm where α is the concentration of Doxycycline drug, β is the coefficient of inhi-bition, γ is degradation coefficient and K is the concentration of x when the5eedback is half of the maximum. Plotting x[t] Vs. time(t)-at d = 16 (Somatic Cell)at d=17 (Pluripotent Cell) ( α =0.5, β =1, γ =0.5,K=1),Initial condition(x[0]=0)Similarly I find ’d’ for Pluripotent Cell upto 5 intermediate cells.I got a linearrelation between d and no.of intermediate cells(n). I plotted it (d in y axis andno. of intermediate states in x axis).and the deviation of the least fit straight line is (y axis)The empirical equation of d and n is= d = 2.4+5n6 .3 Using Doxycycline in One cell what will be the effectof that other cell Here I want to see the effect in other cell due to Doxycycline in one cell. Iconsidered the self regulation and mutual inhibition in this two cells. The math-ematical model is dxdt = α [ d − t ] + β (cid:32) K K + y (cid:33) + δ (cid:32) x K + x (cid:33) − γxdydt = β (cid:32) K K + x (cid:33) + δ (cid:32) y K + y (cid:33) − γy Plot x[t] and y[t](Y axis) vs.time (x axis)(where α is the concentration of Doxycycline drug, β is the coefficient ofinhibition, δ is the coefficient of self regulation and γ is degradation coefficient, α = β = γ =1,,K=1,d=30,y[t]-Orange and x[t]-Blue,initial conditions y[0]=0,x[0]=0) Now,My motivation behind this mathematical model is that there is some bi-ological steps due to chemical reactions which cause delay in regulation andfinding ”Area 51” from this model. I considered the long delay in 2 intermedi-ate states. I changed the previous mathematical model(Here I considered only2 intermediate states only) with multiplying a small constant (I choose (cid:15) = 0.1)to the differential equations corresponding y and z.Actually it’s called forwardfeedback loop,which is more biologically relevant. dxdt = α [ d − t ] + β (cid:16) z K + z (cid:17) − γxdydt = (cid:15) (cid:32) β (cid:16) x K + x (cid:17) − γy (cid:33) dzdt = (cid:15) (cid:32) β (cid:16) y K + y (cid:17) − γz (cid:33) where α is the concentration of Doxycycline drug, β is the coefficient of inhibition,K is the concentration of x when the feedback is half of the maximum and γ is7egradation coefficient.x[t],y[t],z[t] along y axis and time along x axis(d=69.67, α =0.5, β =1, γ =0.5,Initial conditions x[0]=0,y[0]=0,z[0]=0,x is blue,y isorange,z is green).So I find some critical point at d = 69.5 after that the cellwill go to Pluripotent cell. In this paper I gave an mathematical model of Cell Reprogramming from in-termediate differential regulator regulations, which is more biologically rele-vant.This work is supported by Prof. Sitabhra Sinha in IMSc Chennai. Healways helped me in his busy time schedule.I am thankful to IMSc, Chennai forfunding. I used Wolfram Mathematica software for generating plots.6 References