Me abolic Algo i hm wi h Time- a ying Reac ion
Maps
Luca Bianco, Fede ico Fon ana, Vincenzo Manca
Uni e si y o Ve ona
Depa men o Compu e Science
s ada Le G azie, 15
37134 Ve ona, I aly
E-mail: {bianco, on ana}@sci.uni .i ,[email p o ec ed]
Summa y. A symbolic-based app oach o modelling biochemical p ocesses and cellula
dynamics is likely o u n use ul in compu a ional biology, whe e a emp s o ep esen
he cell as a huge, complex dynamic sys em mus ade wi h he linguis ic na u e o he
DNA and he indi idual beha io o he o ganelles li ing wi hin. The ea ly e sion o
he me abolic algo i hm ga e a i s answe o he p oblem o ep esen ing oscilla o y
biological phenomena, so a being ea ed wi h adi ional (di e en ial) ma hema ical
ools, in e ms o ew i ing sys ems. We a e now wo king on a u he e sion o his
algo i hm, in which he ule applica ion is uned by eac ion maps depending on he
speci ic phenomenon unde conside a ion. Success ul simula ions o he B ussela o , he
Lo ka-Vol e a popula ion dynamics and he PKC ac i a ion os e po en ial applica ions
o he algo i hm in sys ems biology.
1 In oduc ion
Symbolic ew i ing has adi ionally been used o s udy and classi y o mal lan-
guages [18]. I was some yea s a e Chomsky’s undamen al disco e ies ha ew i -
ing sys ems began o be applied o he s udy o he g ow h o some simple o gan-
isms and o he analysis o biological s uc u es [9, 15].
These ea ly applica ions o ew i ing o p ac ical case s udies aken om he
eal wo ld demons a ed he po en ial abili y o a p ope ly de ined o mal cons uc
o ep esen , in p inciple, he de elopmen o a leas some biological species. Such
cons uc s, in ac , mo e s ep by s ep owa d he de ini ion o a language/s uc u e
un il hei compu a ion e mina es, hence hei applica ion o species in de elop-
men emphasized he possibili y o o mal sys ems o igu e ou no only classes
o languages, bu also he pa hs along which hei inal s uc u e akes o m du ing
he sys em e olu ion.
Recen ly, a esea ch line has s a ed which ocuses on he ew i ing sys em
dynamic ac i i y ins ead o i s exp essi e powe e alua ed in e ms o language
44 L. Bianco, F. Fon ana, V. Manca
ypes [20, 3, 11]. This line has been s imula ed in an a emp o cap u e, by
means o ew i ing sys ems, he dynamics o a biochemical p ocess. In his a emp
a no el cons uc known as P sys em has come use ul, p o ided i s capabili y
o ep esen se e al s uc u al aspec s o he cell along wi h many in a- and
ex a-cellula communica ion mechanisms [16, 4]. Such a dynamic pe spec i e on
ew i ing employing P sys ems has al eady led o al e na i e ep esen a ions o
di e en biological dynamics [1, 19] and o new models o impo an pa hological
p ocesses [3, 14].
In [3, 11] we ha e s a ed o de elop a me abolic algo i hm ha in oduced a
new pe spec i e in he ew i ing mechanism o P sys ems:
1. ules a e no applied o objec s. Ra he , hey a e applied o popula ions o
objec s;
2. ules a e speci ied along wi h eac i i ies. E e y eac i i y deno es he abili y
o he co esponding ule o compe e agains o he ules in cap u ing pa o
a popula ion, on which he eac ion is pe o med.
We ha e gone u he in his pe spec i e, by associa ing e e y eac i i y o a
map ha depends on he s a e o he sys em. Mo eo e we ha e added a s a egy
o pa i ioning he objec s in he sys em a e e y ansi ion, depending on he
ela i e magni ude o e e y eac i i y.
The pe o mance shown by he me abolic algo i hm in he simula ion o well-
known biochemical models, such as he Lo ka-Vol e a popula ion dynamics [10,
21], he BZ chemical eac ion [5], and he PKC ac i a ion p ocess [2], os e s
po en ial applica ions in c i ical open p oblems deal wi h by sys ems biology
[7]. Simula ion in p og ess a e con i ming he e ec i eness o he algo i hm in
modelling e en mo e complex biochemical p ocesses, such as hose ha e oke
ci cadian hy hms in li ing bodies [8].
2 Me abolic Algo i hm
As we ha e old in he in oduc ion, he me abolic algo i hm is buil on P sys ems.
Fo he sake o simplici y he e, and in he ollowing, we hypo hesize ha his
sys em is made o jus one memb ane. In Sec ion 5 we will b ie ly discuss he
o mal ex ensions needed o cope wi h mo e memb anes.
To p o ide ou algo i hm wi h lexibili y we will gua an ee he ul illmen o
he wo ollowing p inciples a any ansi ion o a P sys em Π, wo king on he
alphabe A={X, Y, . . . , Z}and p o ided wi h ules , s, . . . , w ∈R:
•inc easing he ac i i y o a ule implies a p opo ional dec ease o he ew i ing
ac i i y o o he ules sha ing he same symbols. This condi ion e lec s he
concu ency among ules o e a ini e se o objec s in he sys em;
• he applicabili y o ules is limi ed by hose objec s, whose a ailabili y in he
sys em is low. This condi ion e lec s a cons ain on esou ces.
Me abolic Algo i hm wi h Time- a ying Reac ion Maps 45
In he ollowing we will e o mula e hese wo poin s in quan i a i e e ms.
In he ea ly e sion o he me abolic algo i hm we had pos ula ed ha p ope
eac i i y cons an s a ec ed he ew i ing ac i i y o e e y ule, espec i ely, in a
way ha a la ge eac i i y cons an de ined a highe ew i ing ac i i y o he co -
esponding ule. Then, his ac i i y was p ope ly limi ed by de ining a popula ion
on which a ule :α →β , ans o ming he s ing α ∈ A∗in o a new s ing
β ∈ A∗, could be applied du ing a ansi ion depending on he numbe o objec s
a ailable in he sys em immedia ely be o e ha ansi ion.
The new e sion o he me abolic algo i hm equi es, i s ly, o ecognize he
s a e o he sys em. This s a e is used inside so-called eac ion maps which gen-
e alize he o me eac i i y cons an s in o ime- a ying unc ions. Once we ha e
such maps a hand we will le he ules wo k acco ding o he ela i e eac i i y
exp essed by e e y map, meanwhile limi ing his powe o a oid o e -consump ion
o he objec s in he sys em. Finally, a simple s ochas ic me hod will be adop ed
o decide how o ea indi idual objec s in he sys em, o which he p ocedu e
desc ibed so a does no ake a de ini e decision. Mo e in gene al, in oducing
s ochas ic p ope ies in an algo i hm can u n ou o be pa icula ly desi able
when he dynamics is highly in luenced by ew molecules [6].
2.1 S a e o he Sys em
In classical dynamic sys ems he alues assumed along ime by e e y a iable
usually o m he s a e o he sys em. In a simila way, he e we pos ula e ha a
e e y disc e e ime he numbe o objec s o each ype is well de ined o e e y
memb ane. Fo mally, he s a e o he sys em a ime is iden i ied by a unc ion
q :A −→ N,(1)
whe e Ais he alphabe o he P sys em.
Fo ins ance, q (X) gi es he amoun o objec s Xa ailable in he sys em a
ime . No e ha we will usually omi o deno e , excep o hose cases in which
speci ying he ime u ns ou o be con enien .
The s a e, hence, can be ead by applying q o e e y symbol o he alphabe .
The se o all s a es assumed along ime by he sys em is exp essed by:
Q={q | ∈N}.(2)
This se , hen, con ains he comple e in o ma ion on he e olu ion o he sys em.
Fu he insigh on he s a e is no possible since q is he only p obe we can use
o obse e i .
2.2 Reac ion Maps
As opposi e o he ea ly me abolic algo i hm [3] in which he eac i i y cons an s
had a di ec (and ime in a ian ) ole on ew i ing, he e we gene alize such con-
s an s in o eac ion maps, one o each ule, in a way ha e e y eac ion map gi es
46 L. Bianco, F. Fon ana, V. Manca
he eac i i y ha he co esponding ule has when he sys em is in a gi en s a e.
I ollows ha such maps a e ime a ying, i.e., hey in gene al speci y di e en
eac i i ies in co espondence o di e en empo al s eps.
Fo mally speaking, o each ule we de ine a eac ion map F ha maps s a es
in o eal numbe s:
F :Q−→ R.(3)
Since qis de ined a any empo al s ep, he applica ion o a eac ion map F
ul ima ely esul s in a posi i e eal numbe ha we will ake as he eac i i y o
in q.
Such maps allow o a wide choice o possible de ini ions depending on he
biological phenomenon unde analysis: acco ding o he adi ional o mula ion o
dynamic sys em i is no es ic i e o conside eal unc ions ha in hei s uc-
u e include he s a e o he sys em plus ac o s such as he eac i i y cons an s
men ioned a he beginning o his sec ion.
As an example, conside a ew i ing sys em ha ing an alphabe made o i e
symbols, A={A, B, C, D, E}, and wo ules, and s:
:ABB k
→AC
s:AE ks
→BD (4)
in which, consis en ly wi h no a ions adi ionally adop ed in biochemis y, we
ha e speci ied cons an eac i i ies (k and ks) ha a e peculia o each ule—
hey could be, o example, kine ic pa ame e s ela ed o he chemical eac ions
espec i ely associa ed o hese ules.
Possible s uc u es o he eac ion maps migh be he ollowing ones:
•simple eac i i y cons an s
F =k
Fs=ks
• eac i i ies d i en by he law o mass ac ion
F =k q(A)q(B)
Fs=ksq(A)q(E)
• eac i i ies depending on only he la ges numbe o objec s in he sys em ha
a e isible o he ule
F = max{q(A), q(B)}
Fs= max{q(A), q(E)}
• eac i i ies depending on an ex e nal p omo e , like an enzyme capable o ac-
i a ing he eac ion
F =q(D)
Fs={q(D)}2
Me abolic Algo i hm wi h Time- a ying Reac ion Maps 47
In he ollowing we will pick up his example as long as we need o illus a e
he p inciples o he algo i hm.
2.3 Reac ion Weigh s
Reac ion maps a e no used di ec ly, as eac i i ies. Ra he , hei ac i i y is p o-
po ionally dis ibu ed among he ules by means o so-called eac ion weigh s.
E e y eac ion weigh hen gi es, o each symbol, a popula ion amoun a ule ap-
plies o in o de o p opo ionally consume he co esponding objec . By deno ing
wi h α(i) he i h symbol in a s ing α, wi h |α| he leng h o he same s ing, and
wi h |α|X he numbe o occu ences o Xin α, hen we de ine he eac ion weigh
W ¡α (i)¢ o :α →β wi h espec o he symbol α (i).
No maliza ion can be s aigh o wa dly exp essed in quan i a i e e ms i we
hink ha all ules co-ope a e, each one wi h i s own eac i i y, o consume all
a ailable objec s. Thus, i mus be:
X
ρ∈R|X∈αρ
Wρ¡X¢= 1 ∀X∈ A (5)
ha is, o each symbol he sum o he eac ion weigh s made o e he ules
con aining ha symbol in hei le pa equals uni y.
Holding his cons ain , we can de ine he eac ion weigh s o each ∈Ras
W ¡α (i)¢=F
X
ρ∈R|α (i)∈αρ
Fρ
, i = 1, . . . , |α |(6)
No e ha , simila ly o wha happens in (5), we sum a he denomina o o e he
ules con aining he symbol α (i) in hei le pa .
Re u ning o ou example, we ha e o compu e W (A), W (B), Ws(A), Ws(E):
W (A) = F
F +FsW (B) = F
F = 1
Ws(A) = Fs
F +FsWs(E) = Fs
Fs= 1
(7)
2.4 Limi a ion, Rounding and S a e T ansi ion
Fo wha we ha e said in he abo e, he a ailable objec s a e consumed p opo -
ionally o he eac ion weigh s. Then, in ou example we ha e o choose whe he
o consume
W (A)q(A) o W (B)q(B) (8)
objec s using , and
Ws(A)q(A) o Ws(E)q(E) (9)

48 L. Bianco, F. Fon ana, V. Manca
using s—p o ided o simplici y ha all alues in (8) and (9) a e in ege .
The igh choice is igu ed ou by conside ing ha e e y ule canno consume
mo e han he amoun o he ( eac an ) objec , aken wi h i s own mul iplici y in
he eac ion, whose a ailabili y in he sys em is lowes . Limi a ion, hen, comes
ou o e e y ule by minimizing among all eac an s pa icipa ing o i :
Λ = min
i=1,...,|α |nW ¡α (i)¢q¡α (i)¢
¯¯α ¯¯α (i)o(10)
S ill, Λ is a eal numbe . As opposi e o his, a genuine objec -based ew i ing
sys em mus es ic he ule applica ion domain o in ege alues. Ins ead o ,
o ins ance, ounding he minima ob ained by (10), we p e e he ollowing policy
(la e we will unde s and why): o e e y ule, compa e he ac ional pa ac(Λ )
o Λ o a andom a iable de ined be ween 0 and 1, and choose he loo o Λ
i his ac ion is smalle , he ceiling o he wise. In his way, new ounded minima
esul o be equal o:
Λ =½ loo (Λ ), ac(Λ )≤
ceil(Λ ), ac(Λ )>
.(11)
As a esul o his s ep we ob ain he se {Λ , ∈R}, con aining he numbe
o objec s each ule will be applied o.
In conclusion, o e e y symbol X∈ A he change in he numbe o objec s
due o is equal o he s oichiome ic ac o o , equal o |β |X− |α |X, imes
he alue Λ :
∆ (X) = Λ (|β |X− |α |X) (12)
I descends ha o e e y symbol X∈ A he s a e e ol es acco ding o he
ollowing o mula:
q +1(X) = q (X) + X
∈R
∆ (X) (13)
Again in ou example, le us suppose ha a ime i is q(A) = q(B) = q(C) =
q(D) = q(E) = ˜q, u he mo e F = 3/4Fs. Then,
Λ = minn3/4Fs
3/4Fs+Fs
˜q, 1
2˜qo=3
7˜q
Λs= minnFs
3/4Fs+Fs
˜q, ˜qo=4
7˜q
A e ounding Λ and Λs(he e, o simplici y, we suppose o ha e ound in ege s
al eady a he limi a ion s ep) we ha e
∆ (A) = 3
7˜q(1 −1) = 0
∆ (B) = 3
7˜q(0 −2) = −6
7˜q
Me abolic Algo i hm wi h Time- a ying Reac ion Maps 49
∆ (C) = 3
7˜q(1 −0) = 3
7˜q
∆s(A) = 4
7˜q(0 −1) = −4
7˜q
∆s(B) = 4
7˜q(1 −0) = 4
7˜q
∆s(D) = 4
7˜q(1 −0) = 4
7˜q
∆s(E) = 4
7˜q(0 −1) = −4
7˜q
in a way ha
q +1(A) = ˜q−4
7˜q=3
7˜q
q +1(B) = ˜q+³−6
7+4
7´˜q=5
7˜q
q +1(C) = ˜q+3
7˜q=10
7˜q
q +1(D) = ˜q+4
7˜q=11
7˜q
q +1(E) = ˜q−4
7˜q=3
7˜q
F om he las equa ions i ollows ha
X
X∈A
q +1(X) = 32/7 ˜q < X
X∈A
q (X) = 5˜q
In e es ing o see, in his sys em he o al numbe o objec s canno inc ease along
ime. In o he wo ds in ou example he ollowing ela ion holds:
X
X∈A
q +1(X)≤X
X∈A
q (X)
3 Flexibili y o he Algo i hm
The p oposed algo i hm has wo basic access poin s whe e pa ame e s can be pu
in o: he eac ion maps, and he s ochas ic p ope ies o .
•Reac ion maps can be de ined wi h ela i e eedom, and e en changed du ing
he p ocess acco ding o he speci ic phenomenon unde s udy. Thei ac i i y,
in ac , is in any case no malized by he eac ion weigh s. Occasionally some
maps may esul in null alues: in his case eac ion weigh s migh a ise in he
o m 0/0, and p ope s a egies mus be pu in o ac ion o handle hem p op-
e ly. Reac ion maps, in conclusion, enable he ine con ol o he mac oscopic,
i.e., de e minis ic pa o he p ocess.
50 L. Bianco, F. Fon ana, V. Manca
•Con e sely, he s a is ics o has consequences on he sys em beha io ha
become as mo e impo an , as ewe objec s a e p esen in he sys em. In o he
wo ds i in luences he s ochas ic pa o he p ocess, i.e., i s unp edic abili y in
on o indi idual d i s om he a e age beha io . Al hough u he esea ch
mus be ca ied ou o shi he me abolic algo i hm close o s ochas ic me h-
ods used in biochemis y [17], ne e heless he con ol o al eady allows o
handle, a leas o some ex en , an in e es ing p ope y o mos disc e e popula-
ion dynamics, acco ding o which he decision aken by an indi idual becomes
as mo e c ucial, as less popula ed he sys em is [6]. This ea u e is e iden in
he simula ion o he Lo ka-Vol e a dynamics p oposed in he ollowing.
The ounding policy exp essed by (11) does no p e en ha he esul ing
applica ion o ules exceeds he a ailable esou ces in he sys em. As an example
suppose ha , du ing a ansi ion, i happens ha Λ ≥Λ o each ∈R: in
his case i is likely ha he consequen applica ion o he ules o e -consumes
a leas some objec s a ailable in he sys em. To p e en his we mus check
ha X
∈R
Λ |α |X≤q(X)∀X∈ A (14)
o he wise he se o minima mus be compu ed again.
In i s app oxima ion can be chosen o ha e a uni o m dis ibu ion. We
will p esen he e an example in which a di e en choice o he andom a iable
leads o mo e accu a e simula ion esul s.
3.1 T anspa en Rules
The me abolic algo i hm allows o une he ac i i y o ules. Tuning is achie ed by
adding in he sys em so-called anspa en ules, i.e., ules in he o m α →α .
P ope eac ion maps can be selec ed o pu such anspa en ules in concu ence
wi h he o he , e ec i e ules sha ing common eac an s. In his way, du ing a
ansi ion o he sys em e e y ule is applied as less in ensi ely, as la ge he
eac i i y alue exp essed by a concu en anspa en ule is.
Fo ins ance, le us add a ule :Ak
→Ain he sys em exp essed by (4). This
leads o he ollowing eac ion weigh s:
W (A) = F
F +Fs+F W (B) = F
F = 1
Ws(A) = Fs
F +Fs+F Ws(E) = Fs
Fs= 1
(15)
No e ha we can omi o compu e W (A) due o he anspa ency o he co e-
sponding ule.
Clea ly, F unes he ac ion o and so e he symbol A. In he limi case
F =∞ he ule inhibi s he ac ion o and so e A, since in his case we ha e
W (A) = Ws(A) = 0. Finally, a ule :ABE →ABE would inhibi bo h and s
in he same limi case.
Me abolic Algo i hm wi h Time- a ying Reac ion Maps 51
T anspa en ules add u he lexibili y o he algo i hm. In pa icula , hey
allow o obse e he sys em e olu ion wi h he desi ed deg ee o esolu ion e-
ga dless o any conside a ion abou he g anula i y o he empo al s ep. Changes
in esolu ion a e ins ead ob ained by “hiding” objec s o he sys em e olu ion by
means o anspa en ules. The way anspa en ules wo k e lec s an inhe en
a i ude o he me abolic algo i hm o scale i s own esolu ion no along he ime
dimension, i.e., by means o a empo al scaling ac o as i happens in mos nu-
me ical me hods. Ra he , esolu ion is scaled by adap ing he size o popula ions
o he deg ee o p ecision expec ed o he expe imen .
4 Resul s
We show esul s coming om he p eda o -p ey popula ion dynamics, he B usse-
la o , and he PKC ac i a ion p ocess.
4.1 P eda o -P ey Popula ion Dynamics
The classic Lo ka-Vol e a popula ion dynamics [10, 21] can be desc ibed by a
simple se o ew i ing ules in which Xa e p eys and Yp eda o s:
:Xk
−→ XX p ey ep oduc ion
s:XY ks
−→ Y Y p eda o ep oduc ion
:Yk
−→ λp eda o dea h
(16)
He e, we can une he ac i i y o e e y ule by selec ing p ope eac i i y cons an s
k ,ksand k p opo ional o he a e o ep oduc ion and dea h o bo h p eda o s
and p eys. We pos ula e Fs o be cons an ly p opo ional o ks imes he maximum
numbe be ween p eys and p eda o s, max{q(X), q(Y)}, ha a e p esen in he
popula ion a any sys em ansi ion. Con e sely, he emaining eac ion maps a e
se o be cons an ly equal o he co esponding eac i i y cons an s:
F =k
Fs=ksmax{q(X), q(Y)}(17)
F =k
Mo eo e we add anspa en ules accoun ing o p eys ha a e no ep oduc-
ing o being consumed and o p eda o s ha a e no ea ing o dyeing:
u:Xku
−→ Xp ey s anding by
:Yk
−→ Yp eda o s anding by (18)
The se o eac ion weigh s is hen equal o
58 L. Bianco, F. Fon ana, V. Manca
0 5 10 15 20 25 30
0
10
20
30
40
50
60
70
80
90
100
s ep
concen a ion
Ca
PKC−a
DAG
PKC−i
Fig. 4. PKC ac i a ion dynamics. The o de o he elemen s in he legend is he same
as he o de o hei inal concen a ions wi hin he plo .
meanwhile we associa e a cons an eac i i y map o each anspa en ule, in ou
case F= 50. These eac i i y maps a e qui e simple bu in he u u e we in end
o in es iga e he e ec i eness o mo e complex eac i i y maps in he case o he
PKC model.
The desc ip ion o he whole se o weigh s W i, ha can be calcula ed ob-
jec by objec in he way in oduced in p e ious sec ions, is omi ed. Ra he , we
p esen some simula ion esul s ob ained using ou algo i hm. In Figu e 4 we see
ha , in acco dance wi h esul s ob ained in [2], PKC −idec eases o ze o while
PKC −ag ows up un il eaching a s a iona y maximum. Figu e 5 ep esen s he
cha ac e is ic dynamics o he diacylglyce ol-p o ein kinase C (DAG.PKC) com-
plex.
5 Discussion
All ules discussed so a do no p esen any a ge speci ica ion. This aspec needs
u he discussion due o i s impo ance.
Le ’s conside he ollowing ule , p esen in a memb ane wi:

Me abolic Algo i hm wi h Time- a ying Reac ion Maps 59
0 10 20 30 40 50 60 70 80 90 100
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
s ep
concen a ion
DAGPKC
Fig. 5. DAG.PKC complex dynamics.
AB →BINjC , F
whe e F is he eac i i y map associa ed o . I s meaning is he ollowing: when-
e e Ajoins Binside wi, hey combine and p oduce an objec Cinside he same
memb ane, meanwhile an objec Blea es wiand eaches he memb ane wj. In
such a way a ec s objec s ha a e p esen in wo di e en memb anes. In pa -
icula , om a s uc u al iewpoin , he elemen s B ha a e p esen in wiha e
o be dis inguished om he elemen s B ha a e p esen in wj(and, in ac , his
is he e ec o compa men aliza ion). Fo his eason o igina es ou me abolic
equa ions desc ibing he beha io o i s ou dis inc elemen s:
∆ (Awi) = −Λ ,wi
∆ (Bwi) = −Λ ,wi
∆ (Bwj)=+Λ ,wi
∆ (Cwi) = +Λ ,wi
whe e we ha e in oduced he label o he memb ane con aining e e y elemen as
subsc ip .
In his way we can see ha he a ia ion due o on he objec s Bplaced
inside wj, i.e., ∆ (Bwj), depends on he concen a ions o Aand Bloca ed in wi
60 L. Bianco, F. Fon ana, V. Manca
as s a ed by he subsc ip no a ion (no e ha his dependence is hidden behind he
Λ ac o ). This simple e olu ion ule is powe ul enough o show ha mo emen s
o objec s be ween memb anes can be handled easily by conside ing, as dis inc
elemen s, wo objec s o he same ype loca ed in di e en egions. This addi ional
in o ma ion in oduces a no a ional o e head o a ge ing e e y objec wi h he
label o he memb ane con aining he espec i e objec . On he o he hand i does
no in oduce any concep ual complica ion.
The case in which elemen s appea ing in he an eceden o he ule a e placed
inside di e en memb anes can be handled simila ly. The only di e ence ha mus
be aken in o accoun is ha he se o weigh s has o be calcula ed by conside ing,
in p inciple, he whole se o ules o he P sys em a he han he se o ules o
a single memb ane.
6 Conclusion and Ongoing Resea ch
Sys ems biology demands o no el p ocedu es capable o ep esen ing biological
p ocesses wi h bo h accu acy and lexibili y. In on o a huge and well- oo ed
amily o nume ical schemes, adi ionally de o ed o igu e ou he dynamics
o sys ems desc ibed by di e en ial equa ions, al e na i e algo i hms based on a
symbolic ep esen a ion o he phenomena p omise o deal mo e na u ally wi h
he s uc u al cha ac e is ics o he biomolecules and wi h he biochemical eac-
ions such molecules gi e ise o. By using he same kind o ep esen a ion, ou
algo i hm mo eo e seems o handle in a s aigh o wa d and e icien way hose
condi ions in which ew molecules ha e an impo an impac on he sys em dy-
namics, whe e mos adi ional nume ical s a egies a e no longe e ec i e and
mus be subs i u ed by s ochas ic algo i hms.
Success ul simula ions conduc ed on wo pa adigma ic nonlinea p ocesses in
biochemis y, namely he Lo ka-Vol e a popula ion dynamics and he BZ eac-
ion, plus he expe imen conduc ed wi h he PKC ac i a ion p ocess, ask o
u he es he po en ial o he me abolic algo i hm. Ou p esen esea ch aims o
simula e some undamen al signal ansduc ion ne wo ks, in pa icula he PER
and TIM cycle in he ci cadian oscilla ion in D osophila.
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