11
Theory of genes network in reprogramming of iPS cells
Kazuhisa Nishi

School of Engineering,
University of Hyogo
1. Introduction

Reprogramming of somatic ceıls into iPS cells can be achieved by expression of severaltranscriptional factors (Takahashi and Yamanaka, 2006; Takahashi et al., 2007; Okita et al.,2007; Wemig et al., 2007; Maherali et al., 2007; Yu et al., 2007). There are some distinct
properties in the reprogramming of iPS celıs compared with nuclear transfer or cell fusion insomatic cells (Gurdon et al., 1958; Wilmut et al., 1997; Tada et al., 2001; Cowan et al., 2005).
The first is that the specified transgenes (Oct4, Sox2 etc.) are needed in reprogramming, thesecond is that the endogenous loci of the transgenes should be expressed, and the third is that
iPS celı generation requires several cell division. These facts imply that if the genes
transcriptional network of somatic cells is activated by the specific factors, they could befinally reprogrammed by changing the total expressing pattem of genome. However, its
mechanism still remains unsolved probıem. In this paper I suggest a reprogramming
mechanism of iPS ceıls, emphasizing how the dynamics of genes transcriptionaı network canbe related with deterministic and stochastic processes.
2. Equations of gene network

The discovery of the iPS ceıls leads to the astonishing facts that tranceducing trigger genes
into somatic cells can change the expressing pattems of genome drastically. In order to findthe reprogramming mechanism of iPS cels, we wilı have to consider not only the genes
network of the specified factors such as Sox2, Oct3 etc but also the total dynamics of genome
with interacting between these pluripotency relevant factors. First ıet us consider basic
equations used in the genes networks. Figure 1 shows the model of transcription factors
network in iPS cells which is the positive feedback co‐operated with Oct4, Sox2, and Nanog.
Extemally tranceduced Oct4 and Sox2 activate Oct4 and Sox2 enhancer, which expresses
endogenous Oct4, Sox2, Nanog and so on. Further, they begin to activate Oct4‐Sox2 enhanceL
and a basic network of iPS cells is finally stabilized in the expression ofthe endogenous genes,
Mathematicaı models of the gene‐network have been almost applied for the system in which
Basic NetworkFig. 1. Genes transcription network constructed with pluripotency related genes and proteins.

2
the genes are directly interacting with transcription factors (Elowitz and Leibler, 2000;
Gadner et al., 2000). However, since in the eukaryote cells, genes are constituted with thechromatin structures, the control system of the gene transcription is not the same as the caseof the prokaryote cells. Nevertheless, it is for simpıicity assumed here that the main frame of
the model equations adopted in the prokaryote cells can be also applied for the case of theeukaryote cells. To carry out this treatment efficientıy, the system parameters or coefficients inequations w ıl be re‐defined so as to reflect the interaction between the transcriptionaı factors
and the chromatin structures. The network starts with a mathematical model of transcriptional
regulation, which is represented by Michaelis‐Menten equation that describes the chemical
reaction of the feedforward control constituted with Oct4, Sox2 and Nanog. It is consideredthat Oct4‐Sox2 enhancer is co‐operatively activated by the product of the Oct4 and Sox2
expressing quantity.
Basic equations ofthe system are given by\cdot\frac{dm_{oc(x)} {dt}=-m_{x(sx)}+\frac{\alpha p_{oc}p_{\alpha} { \imath}+ p_{oc}p_{x} , \frac{dp_{oc(\alpha)} {dt}=-\beta(p_{oc\langle\alpha)}-m_{oc(x) })\frac{dm_{ng}}{dt}=-m_{ng}+\frac{ap_{ed}}{1+p_{eA}}, \frac{dp_{ng}}{dt}=-\beta (p_{ng}-m_{ng})(1).

where the variable p_{oc}, p_{sx},p_{ng} are Nanog, Oct4, Sox2 protein number per cell in units of K_{m}(the number ofproteins necessary to half‐maximally activate a promoter), respectively, andm_{oc}, m_{sx}, m_{ng} are the corresponding mRNA
respectively.
It shows whether thenumber per cell rescaled by the average
expression of Oct4 or Sox2 can be stabilized
number of proteins produced per mRNAin a cell or not is determined by the initialmolecule. The coefficient ctı, \alpha_{2},\alpha_{3} arequantity ofthese proteins.
maximum numbers of proteins per cell ofNanog, Oct4, Sox2, respectiveıy. The16
coefficient \beta is the ratio of the protein decayf^{t5}
rate to the mRNA decay rate. The action of\backsola\ohv\aolvbaolbxo{x\{\tt\ssmmalalREREJJ CTE}CT4=\}P3hi
the network depends on several factors,
8^{12}
including the dependence of transcription
rate on activator or repressor concentration,
the translation rate, and the decay values ofC39
the protein and the messenger RNA. Figure\Phi\subseteq 8
2 shows that Oct4 or Sox2 proteins number\dot{8}7
per ceıl in units of K_{m} (the number of\varepsilon\triangle ft\alpha6
proteins necessary to half‐maximally\infty ox5
activate a promoter) is represented as a\dot{o}4D\in 10v^{1 }function of time rescaled in units of themRNA decay lifetime. Here, \alpha ı + a2, a_{3} aremaximum numbers of proteins per ceıl ofOct4 or Sox2 and Nanog, respectively, and \betais the ratio of the protein decay rate to themRNA decay rate. The values of p_{1}, p_{2}, p_{3}indicate the initial numbers of these proteins,\backslash \vee 3

g\dot{o}2ot

0T2.

\ovalbox{\t \smal REJECT} me[t/1fetime4.6 mRNA]81ofFig. 2. Oct3/4 or Sox2 protein number.

103Effective deterministic equations are defmed as the formuıa of input and output constructedwith only basic equations.

\frac{dm_{mt} {dt}=-m_{out}+\frac{\alpha p_{ln} {1+p_{m} , \frac{dp_{\alpha t} }{dt}=-\beta(p_{\alpha t}-m_{wt}).

(2)Further, stochastic connection of deterministic equations are also defined as

3. Reprogramming process

Now let us consider the reprogramming process of somatic cells. In order to understand thereprogramming mechanism, the total dynamics of genome has to been revealed. First threegroups of cells which are somatic, partially reprogrammed, and iPS cells wiıl be considered
and be factorized into G1\sim G5 groups, shown in the left side of Fig. 3. The right side of Fig. 3
shows the process model indicating how the expression pattems of genome can change fromsomatic state to reprogrammed state as the result of the interaction between reprogrammed
related genes, master genes of G1\sim G5 groups and other genes as environment.
It is well known that requiring several cell divisions in reprogramming can be one reason
why iPS cell generation is so slow and inefficient. These experimental facts lead us to theconjecture that the wild types of somatic cells can repair the epigenetic modifications aftercell division, but the somewhat artificial ceıls modified by the basic network cannot
sufficiently repair them. The epigenetic modifications of eukaryote cells can be also
Transti\propto’ Process

Genes;()1So\not\subset , Od4, Nanog(i) Master genes of G145(ii) \alpha kr genes asenvironment

genes are actng \varpic\sigmaOctkSox2

in Q of somatic state.

*\backsla h adivted in

Qa/p\mu e\approx\infty roe\dot{n} Gl.

; in G3 are activated by suppression
ne genes in G\uparrow.
enes in G are s\iotappraesd by activation
genes \dot{n} G3.

mes in G5 (Nanog et_{0}) are activated by
ppression of Q

mes \dot{n}\infty are s\iota \mathfrak{p}praesed by activabOn
G5.

rog arming stateFig. 3. left: Genes group G1\sim G5 , right: Process model ofreprogramming mechanism.

4interpreted to be evolutionarily conserved, and this situation may not be applicable for theartificial cells such as iPS/ES . With these considerations on mind, several conditions wiıl beconsidered in the reprogramming process.
There are epigenetic states such as DNA methylation and histon modificstion. Since thereprogramming transition from these epigenetic states has two properties ofactive and passiveprocesses. Then, let us consider the two stochastic processes which are in the absence andpresence of cell divisions. In the absence of cell divisions incıuding both active and passive
reprogramming processes, stochastic reıation between deterministic genes network equations
is given by
P=p_{1}\cross\cdots\cross p_{I}(3)where i shows the number of effective deterministic equations. Next let us consider in thepresence of cell divisions (passive reprogramming process). Given L the number of genes
epigenetically modified in somatic cells. Here r is defined as the probability attaining thereprogrammed state after one cell division per one gene. The expectation value Q_{1} of thegenes in a single DNA, which are in the reprogrammed state after one cell division, on theaverage, is given by

Q_{1}= \sum_{k-1}^{L}Ckr^{k}(1-r)^{L-k}\equiv Lr .

(4)

After one cell division, the number of cells which have the partial reprogrammed region isgiven by

N_{{\imath}}=p_{1}N.

(5)where N is the number of imtial somatic cells and p_{1} is the probability defined by the relation(3). This means the existence of Nı cells which have reprogrammed region Qı. The value Q_{n}after n times celı divisions and the number N_{n} ofpartially reprogrammed cells are representedas

Q_{n}=r(L-Q_{n-1})+Q_{n-1}\equiv L[1-(1-r)^{n}], N_{n}=p_{I}\cross\cdots p,N .

(6)

The reprogrammed ratio in a cell is represented as a function of cell divisions and is shown inFig. 4a.

R_{n}= \frac{\partial}{\partial n}(\frac{Q_{n} {L})=(1-r)^{n}\ln\frac{1}{1-r}
The minimum cell divisions

M for

the completeiPS.

(7)cells is obtained by the following

relations

NPL(1-r)^{M-{\imath}}\geq 1, NPL(1-r)^{M}<1 .

(8)Then, M is given by (shown in Fig. 4b)
M= ı‐\frac{\logNPL}{\log({\imath}-r)} .

(9)

The efficiency E_{f} producing iPS ceıls is also given by (shown in Fig. 4c)

E_{f}=PL(1-r)^{M-1} .

(10)where P=p_{1}\cross\cdots\cross p_{l} . The unknown parameters of PL and r determining the reprogramming

5
processes can be caıculated from equations(7), (9), (10). Choosing the values of R_{1}=0.08, M=50 and E_{f}=1\% which is reaıisticin the experimentaı data, the values ofparameters are evaluated as P=0.001, L=300 and r=0.08 . Let us interpret briefly
some experimental facts related to thesuccess of reprogramming and its producing
efficiency. Why Klfi4 and c ‐Myc enhance theiPS producing can be interpreted by two
possible mechanisms. One of these is todecrease
reprogramming
timebyacceıerating the cell division rate, and theother is to prevent the epigeneticmodifications by the expressing factors ofthese genes. It is well known that theinhibition of p53/p2 pathway does notchange necessary cell divisions to produce

0,(a)_{0.0}0.0\propto_{0.0}^{*}Qr^{0.0}\Phic\overline{E}0.0\varpiEb^{0.0}\l corner 0oe\Phi a000.00. 02004060
celldivisions, n80

Fig. 4 (a) Reprogramming rate0.

(b) 1(c)s\iota\check{u}^{0.0}0.0\underli{oc\dt mega}\hat{\munhbd{Ee}_rol\ienpsieln{c0}}.u\f^ra{c{0o}.00.}w>=_{o}\sim03^{o}0.0\equivn3\i!_{-0. }^{t\hslah}\cir 0. e\equivo.o0

0. 10.20.30.40.50.60.70.80.9

Reprogramming probability per gene, rI00. 02

0. 04

0. 06

0. 08

0.\iota

Reprogramming probability per gene. r

Fig. 4 (b) Minimum cell dvision for reprogramming; (c) iPS producing efficiency
iPS cells, but it decreases the necessary time to produce iPS cells (Hanna et al., 2009). This
corresponds to the case that M is unchanged and \tau is smaller in the model parameters. WhenNanog, that is one of proteins expressing in the basic network, increases, it will increase thevalue of r because the repairing of epigenetic modification is prevented by operation of thebasic network. This explains the experimental fact that Nanog overexpression decreases thenecessary time to produce iPS cells. Only a small portion of the transduced cells become
pluripotent and the others become almost partially reprogrammed state. This phenomenon
may be corresponding to the case of r<<1 in the model. Also, the partially reprogrammed
cells not only express the genes expressing in both somatic and iPS cells, but also express thegenes non‐expressing in these cells (Mikkelsen at al., 2006) In this model, the suppressedgenes in somatic celıs can activate by de‐epigenetic process after celı divisions. Of these6genes, there exist the genes to be suppressive in iPS cells. However, there will be some cells
which are not suppressed by basic network. This case is then considered to be the partially
reprogrammed cells.
4. Conclusions

This model can compute the minimum cell divisions for attaining iPS cell and its generating
efficiency, and can expıain various experimental facts which have been by now reported iniPS cells. The reprogramming mechanism considered here assumes the existence of theundefmed determinants or environmental conditions suppressing the repair enzyme for theepigenetic modifications. If suppressing determinants can be identified, it will make theefficiency of producing iPS cells more improved. More speculatively, since genes control
system is not still closed, it wiıı be modified by the interaction with the unknown environment
or material which must be evolutionally determined. Thus, if the genes control system
experiences the environment which can be never realized in wild types of cells, it wiıl bepossible to change the genes control system drastically. This viewpoint may open a newapproach to controı the gene network including reprogramming treatment.
References

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7. J. B. Gurdon, T. R. Elsdale, M. Fischberg, Nature 182, 64 (1958).
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9. M. Tada, et al., Cum. Biol. 11, ı553 (2001).
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School of Engineering
University of Hyogo
Himeji, Hyogo 671‐2201
Japan

E‐mail address: tyus51kzhs@ares.eonet.ne.jp