34
M. SUGITA
where Xlk is the quantity of a metabolite I in a compartment k, Alk and
Blk are in- and outflux of I and are functions of continuous variables
Xtk> X2k' • •. as well as of the binary quantities, gl' g2' ••. [5]. Let us
consider a certain quantity Yk being the function of Xtk> XZk> ..• having a
physiological meaning, and assume that a pulse g" = 1 is generated, when
(2)
where ckis its threshold value, and that this pulse is fed back and affecting the
internal state of the automaton. If, for example, Xtk and X2k are the quantities of co-repressor and apo-repressor, then Yk may be related to the
repressor. In the relation (3.3) of a preceding paper [6] the symbols rand
fer) were used instead ofYk andg". Thus, the internal state of the automaton
may be determined by the following logical function
gk = Gk (gl, gZ, ... ; PI, 12, ... )
(3)
where PI' 12, ... are exogenous digital inputs, e.g. of fertilization, virus
infection, radiation induced mutation. The block diagram of such an
interacting system is given elsewhere [5].
Application of our Method to Cell Division
Our mathematical method can effectively be applied to some biological
problems like development, cytodifferentiation, etc. During cell cycle some
metabolites regulating mitosis or DNA duplication must fluctuate. GOODWIN [1] transferred the ecological relation of prey-predator to oscillating
metabolites. For this purpose, at least two kinds of metabolites, instead
of one, are required. Let x and u be the metabolites regulating mitosis and
DNA replication respectively, then Y and v of the following kinetic equations
may correspond to the predator. For simplicity let us assume the following
form
where
dx
dt = iX (x,y,u,v)gy-k 1 x,
dy
- = {3 (X,y,u, v)gu-k 2 Y,
dt
gu = 1 when x ~ xc,
gM= 0 when x < xc,
gy = 0 wheny ~ Yc,
gy = 1 wheny < Yc,
dv
dt = (j (x,y,u,v)gD-k4 v
gD = 1 when u ~ Uc
gD = 0 when u < Uc
gv = 0 when v ~ Vc
gv = 1 when v < Vc
(4)
(5)
~ = 1 and gD = 1 are signals of mitosis and DNA replication; xc,Yc etc.
are threshold values. iX, {3, y and (j are functions of x,y, u, and v. However,
M. SUGITA
where Xlk is the quantity of a metabolite I in a compartment k, Alk and
Blk are in- and outflux of I and are functions of continuous variables
Xtk> X2k' • •. as well as of the binary quantities, gl' g2' ••. [5]. Let us
consider a certain quantity Yk being the function of Xtk> XZk> ..• having a
physiological meaning, and assume that a pulse g" = 1 is generated, when
(2)
where ckis its threshold value, and that this pulse is fed back and affecting the
internal state of the automaton. If, for example, Xtk and X2k are the quantities of co-repressor and apo-repressor, then Yk may be related to the
repressor. In the relation (3.3) of a preceding paper [6] the symbols rand
fer) were used instead ofYk andg". Thus, the internal state of the automaton
may be determined by the following logical function
gk = Gk (gl, gZ, ... ; PI, 12, ... )
(3)
where PI' 12, ... are exogenous digital inputs, e.g. of fertilization, virus
infection, radiation induced mutation. The block diagram of such an
interacting system is given elsewhere [5].
Application of our Method to Cell Division
Our mathematical method can effectively be applied to some biological
problems like development, cytodifferentiation, etc. During cell cycle some
metabolites regulating mitosis or DNA duplication must fluctuate. GOODWIN [1] transferred the ecological relation of prey-predator to oscillating
metabolites. For this purpose, at least two kinds of metabolites, instead
of one, are required. Let x and u be the metabolites regulating mitosis and
DNA replication respectively, then Y and v of the following kinetic equations
may correspond to the predator. For simplicity let us assume the following
form
where
dx
dt = iX (x,y,u,v)gy-k 1 x,
dy
- = {3 (X,y,u, v)gu-k 2 Y,
dt
gu = 1 when x ~ xc,
gM= 0 when x < xc,
gy = 0 wheny ~ Yc,
gy = 1 wheny < Yc,
dv
dt = (j (x,y,u,v)gD-k4 v
gD = 1 when u ~ Uc
gD = 0 when u < Uc
gv = 0 when v ~ Vc
gv = 1 when v < Vc
(4)
(5)
~ = 1 and gD = 1 are signals of mitosis and DNA replication; xc,Yc etc.
are threshold values. iX, {3, y and (j are functions of x,y, u, and v. However,