Turing Meets Synthetic Biology - 2009.igem.org2009.igem.org/files/poster/IPN-UNAM-Mexico.pdf ·...
1
Turing Meets Synthetic Biology We present a synthetic network that emulates an activator-inhibitor system. Our goal is to show that spatiotemporal structures can be generated by the behavior of a genetic regulatory network. We implement the model by means of several biobricks. We construct a self activating module and correspondingly an inhibitory one. Self-activation dynamics is given by the Las operon, while the inhibitory part is provided by the Lux operon. Quorum sensing and diffusion of AHL provide the reaction-diffusion me- chanism responsible for the formation of Turing patterns. The importance of our work relies on the fact that we show that the action of the morphogenes as originally proposed by Turing is equivalent to the effect of diffusion of chemicals interacting with the synthetic network, which accounts for the reactive part, a possibility implicit in Turing‛s original work in the context of mor- phogenesis of biological patterns. Self-emerging patterns in an activator-inhibitor network. ABSTRACT Based on the design of the biobricks network we tried to model the behavior it would have within a single cell. To do this we constructed the kinetic laws describing the relationships between the components of our network and IPTG and aTc. We modeled the interactions of the AHL‛s with their respective targets as allosteric regulators affecting the production rate of the catalyzing enzymes, thus affecting the syn- thesis of the AHL‛s. This catalysis was assumed to be a first order kinetics depending only on the concen- tration of the catalyst. The diffusion rate of the AHL‛s was reduced to a first order kinetics considering the membrane area of the bacteria and assuming the complete absence of the AHL outside the cell, there- fore depending only on the production within the cell. Luis J. Martínez Lomeli*, Jesús Pérez Juárez*, Daniel I. Aguilar Salvador*, Gilberto Gómez Correa*, José Martínez Lomeli*, Cristian J. Delgado G*, Francisco J. Razo Hernández*, Ileana de la Fuente Colmenares, Alin Acuña Alonzo, Román A. Acosta Díaz, Eniak Hernandez, Pablo Padilla Longoria, Alexander de Luna Fors, Fabiola Ramirez Corona, Arturo Becerra, Francisco Hernández Quiroz, Elías Samra. Turing formally described his proposal with a set of partial differential equations where is possible to re- present the chemical interactions of the morphogens and the way they move over the space. This kind of dynamics are commonly called reaction-diffusion mechanisms, and the equations that describe them are named Reaction-Diffusion equations: Reaction-Diffusion equations In 1972 A. Gierer and H. Meinhardt published their work called “A theory of biological pattern formation” presenting a network based in the interaction of at least two morphogens acting as an activator and an in- hibitor. The main qualitative dynamics of the morphogens is “short range activation, longer range inhibition and a conceptual distinction between effective concentrations of activator and inhibitor, on one hand, and the density of their sources on the other”. The Activator-Inhibitor Activator-Inhibitor dynamics on a single cell Gierer and Meinhardt Kinetics Nondimensional Gierer and Meinhardt Kinetics Spatial Model Estimated Diffusion Constants Experimental Work To do this we considered the molecular weight of the AHL molecules and we estimated the diameter of the molecules taking into account the linear structure of them. From the articles of A. Einstein: “On the Motion Required by the Molecular Kinetic Theory of Heat of Small Particles Suspended in a Stationary Liquid” and “A new determination of molecular dimensions” we used the method to estimate the diffusion coefficients from the diameter of the molecule using the modified friction coefficient of the water in presence of glucose to improve the accuracy as described. We made a simulation with values previously used by J.D. Murray and the previously estimated diffusion coefficients of PAI and AI molecules with the reaction-diffusion equations of the Gierer and Meinhardt model. The simulation was made in a circular area with zero flux boundary conditions and all the parameters values found in the literature. Afterward we described as ordinary differential equations the time change of the concentrations of PAI and AI based on the kinetic law equations, being “X” PAI and “Y” AI. With this system of differential equations describing the kinetic basis of the reaction and diffusion we proceeded to make a qualitative analysis of its behavior. During this months we acomplished the synthesis of 11 new biobricks (protein generators and inverters) which were add to the Registry, we have the 90% of advance of our system completed as follows. We also have preeliminay data confirming the propper functioning of the activator module, as show in the pictures. No IPTG 160 mM IPTG Conclusions and further work The preliminary results show that the activator system works. The inhibitor will be coupled by the end of the year. In order to perform the experiments to show if our biobricks system is able to produce an spa- tiotemporal pattern, we will use a chemical gradient of IPTG and aTc. The modeling results have given us an insight of how the system will respond to particular chemical initial distributions. Besides we also need to complete the modelling part of the project with different perspectives. The use of stochastic approaches should allow us to introduce noise in the system and look how the emergence of bacteria pattern could be affected. We are not so far to conclude if an activator-inhibitor mechanism implemented with biobricks could gene- rate a spatiotemporal turing like pattern. Main references -Turing A. M., "The Chemical Basis of Morphogenesis", Philosophical Transactions of The Royal Society of London, series B, 237:37–72, 1952 -Gierer A. and Meinhardt H., "A theory of biologycal pattern formation", Kybernetik 12:30-39, 1972 -Murray J.D., "Mathematical Biology II", Springer-Verlag Berlin Heidelberg, 3rd edition, 1993 Initial instability damping by the sinuidal initial condition of dynamical system simulation. The system evolution lead to an nonhomogeneous state but an spatial gradient. € d PAI [ ] dt = k c PAI G PAI T R T max F 1 − T d PAI PAI [ ] − D PAI PAI [ ] d AI [ ] dt = k c AI G AI T R T max F 2 − T d AI AI [ ] − D AI AI [ ] F 1 = LasR + PAI [ ] 2 k d 1 2 1 + LasR + PAI [ ] 2 k d 1 2 1 1 + LuxR + AI [ ] 2 k d 2 2 F 2 = LasR + PAI [ ] 2 k d 1 2 1 + LasR + PAI [ ] 2 k d 1 2
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Transcript of Turing Meets Synthetic Biology - 2009.igem.org2009.igem.org/files/poster/IPN-UNAM-Mexico.pdf ·...
Turing M
eets
Synth
etic B
iolo
gy
We p
rese
nt a
syn
theti
c ne
twor
k th
at e
mul
ates
an a
ctiv
ator
-inh
ibit
or s
yste
m. O
ur g
oal is
to
show
that
spa
tiot
em
pora
l st
ruct
ures
can
be g
ene
rate
d b
y th
e b
ehav
ior
of a
gene
tic
regu
lato
ry n
etw
ork.
We im
plem
ent
the m
odel by
mean
s of
seve
ral bio
bri
cks.
We
cons
truc
t a
self
act
ivat
ing
mod
ule a
nd c
orre
spon
din
gly
an inh
ibit
ory
one. S
elf
-act
ivat
ion
dyn
amic
s is
giv
en
by
the L
as o
pero
n,
while t
he inh
ibit
ory
part
is
prov
ided b
y th
e L
ux o
pero
n. Q
uoru
m s
ens
ing
and d
iffu
sion
of
AH
L p
rovi
de t
he r
eac
tion
-dif
fusi
on m
e-
chan
ism
resp
onsi
ble
for
the f
orm
atio
n of
Tur
ing
patt
ern
s. T
he im
port
ance
of
our
wor
k re
lies
on t
he f
act
that
we s
how
that
the
acti
on o
f th
e m
orph
ogene
s as
ori
gina
lly
prop
osed b
y T
urin
g is
equ
ival
ent
to
the e
ffect
of
dif
fusi
on o
f ch
em
ical
s in
tera
ctin
g w
ith
the s
ynth
eti
c ne
twor
k, w
hic
h a
ccou
nts
for
the r
eac
tive
par
t, a
pos
sibilit
y im
plic
it in
Tur
ing‛
s or
igin
al w
ork
in t
he c
onte
xt
of m
or-
phog
ene
sis
of b
iolo
gica
l pa
ttern
s.
Se
lf-e
me
rgin
g p
att
ern
s in
an
ac
tiv
ato
r-in
hib
ito
r n
etw
ork
.
ABSTRACT
Bas
ed o
n th
e d
esi
gn o
f th
e b
iobri
cks
netw
ork
we t
ried t
o m
odel th
e b
ehav
ior
it w
ould
hav
e w
ithin
a s
ingl
e
cell. T
o do
this
we c
onst
ruct
ed t
he k
ineti
c la
ws
desc
ribin
g th
e r
ela
tion
ship
s betw
een
the c
ompo
nent
s of
ou
r ne
twor
k an
d I
PTG
and
aT
c. W
e m
odele
d t
he i
ntera
ctio
ns o
f th
e A
HL‛s
wit
h t
heir
resp
ect
ive t
arge
ts
as a
llos
teri
c re
gula
tors
aff
ect
ing
the p
roduc
tion
rat
e o
f th
e c
atal
yzin
g enz
ymes,
thus
aff
ect
ing
the s
yn-
thesi
s of
the A
HL‛s
. This
cat
alys
is w
as a
ssum
ed t
o be a
fir
st o
rder
kine
tics
depe
ndin
g on
ly o
n th
e c
once
n-tr
atio
n of
the c
atal
yst.
The d
iffu
sion
rat
e o
f th
e A
HL‛s
was
reduc
ed t
o a
firs
t or
der
kine
tics
con
sideri
ng
the m
em
bra
ne a
rea
of t
he b
acte
ria
and a
ssum
ing
the c
ompl
ete
abse
nce o
f th
e A
HL o
utsi
de t
he c
ell, t
here
-fo
re d
epe
ndin
g on
ly o
n th
e p
roduc
tion
wit
hin
the c
ell.
Luis
J. M
art
íne
z Lo
me
li*,
Je
sús
Pé
rez
Juá
rez*
, Da
nie
l I. A
gu
ila
r S
alv
ad
or*
, Gil
be
rto
Gó
me
z C
orr
ea
*, J
osé
Ma
rtín
ez
Lom
eli
*, C
rist
ian
J. D
elg
ad
o G
*, F
ran
cisc
o J
. Ra
zo H
ern
án
de
z*,
Ile
an
a d
e la
Fu
en
te C
olm
en
are
s, A
lin
Acu
ña
Alo
nzo
, Ro
má
n A
. Aco
sta
Día
z, E
nia
k H
ern
an
de
z, P
ab
lo P
ad
illa
Lo
ng
ori
a, A
lex
an
de
r d
e L
un
a F
ors
, Fa
bio
la R
am
ire
z C
oro
na
, Art
uro
Be
cerr
a, F
ran
cisc
o H
ern
án
de
z Q
uir
oz,
Elí
as
Sa
mra
.
Tur
ing
form
ally
desc
ribed h
is p
ropo
sal w
ith a
set
of p
arti
al d
iffe
rent
ial equ
atio
ns w
here
is
poss
ible
to
re-
prese
nt t
he c
hem
ical
int
era
ctio
ns o
f th
e m
orph
ogens
and
the w
ay t
hey
mov
e o
ver
the s
pace
. T
his
kin
d o
f dyn
amic
s ar
e c
omm
only
cal
led r
eac
tion
-dif
fusi
on m
ech
anis
ms,
and
the e
quat
ions
that
desc
ribe t
hem
are
na
med R
eac
tion
-Dif
fusi
on e
quat
ions
:
React
ion-
Diffu
sion
equ
ation
s
In 1
97
2 A
. Gie
rer
and H
. Mein
har
dt
publish
ed t
heir
wor
k ca
lled “
A t
heor
y of
bio
logi
cal p
atte
rn f
orm
atio
n”
prese
ntin
g a
netw
ork
bas
ed in
the int
era
ctio
n of
at
leas
t tw
o m
orph
ogens
act
ing
as a
n ac
tiva
tor
and a
n in
-hib
itor
. The m
ain
qual
itat
ive d
ynam
ics
of t
he m
orph
ogens
is “
shor
t ra
nge a
ctiv
atio
n, lo
nger
rang
e in
hib
itio
n an
d a
con
cept
ual dis
tinc
tion
betw
een
eff
ect
ive c
once
ntra
tion
s of
act
ivat
or a
nd inh
ibit
or, on
one
han
d, an
d
the d
ens
ity
of t
heir
sou
rces
on t
he o
ther”
.
The A
ctivato
r-Inh
ibitor
Act
ivato
r-Inh
ibitor
dyna
mics
on a
single c
ell
Gie
rer
an
d M
ein
ha
rdt
Kin
eti
cs
No
nd
ime
ns
ion
al
Gie
rer
an
d M
ein
ha
rdt
Kin
eti
cs
Spa
tial M
odel
Est
imate
d D
iffu
sion
Con
stant
s
Expe
riment
al W
ork
To
do
this
we c
onsi
dere
d t
he m
olecu
lar
weig
ht
of t
he A
HL m
olecu
les
and w
e e
stim
ated t
he d
iam
ete
r of
th
e m
olecu
les
taki
ng int
o ac
coun
t th
e lin
ear
str
uctu
re o
f th
em
. F
rom
the a
rtic
les
of A
. E
inst
ein
: “O
n th
e
Mot
ion
Requ
ired b
y th
e M
olecu
lar
Kin
eti
c T
heor
y of
Heat
of
Sm
all
Part
icle
s S
uspe
nded i
n a
Sta
tion
ary
Liq
uid”
and “
A n
ew
dete
rmin
atio
n of
mol
ecu
lar
dim
ens
ions
” w
e u
sed t
he m
eth
od t
o est
imat
e t
he d
iffu
sion
co
eff
icie
nts
from
the d
iam
ete
r of
the m
olecu
le u
sing
the m
odif
ied f
rict
ion
coeff
icie
nt o
f th
e w
ater
in
prese
nce o
f gl
ucos
e t
o im
prov
e t
he a
ccur
acy
as d
esc
ribed.
We m
ade a
sim
ulat
ion
wit
h v
alue
s pr
evi
ousl
y us
ed b
y J
.D. M
urra
y an
d t
he p
revi
ousl
y est
imat
ed d
iffu
sion
coe
ffic
ient
s of
PA
I a
nd A
I m
olecu
les
wit
h t
he
reac
tion
-dif
fusi
on e
quat
ions
of
the G
iere
r an
d M
ein
har
dt
mod
el.
The s
imul
atio
n w
as m
ade i
n a
circ
ular
ar
ea
wit
h z
ero
flu
x b
ound
ary
cond
itio
ns a
nd a
ll t
he p
aram
ete
rs v
alue
s fo
und in
the lit
era
ture
.
Aft
erw
ard w
e d
esc
ribed a
s or
din
ary
dif
fere
ntia
l equ
atio
ns t
he t
ime c
han
ge o
f th
e c
once
ntra
tion
s of
PA
I
and A
I bas
ed o
n th
e k
ineti
c la
w e
quat
ions
, bein
g “X
” PA
I an
d “
Y”
AI.
Wit
h t
his
sys
tem
of
dif
fere
ntia
l equ
atio
ns d
esc
ribin
g th
e k
ineti
c bas
is o
f th
e r
eac
tion
and
dif
fusi
on w
e p
roce
eded t
o m
ake a
qua
lita
tive
an
alys
is o
f it
s behav
ior.
Dur
ing
this
mon
ths
we a
com
plis
hed t
he s
ynth
esi
s of
11
new
bio
bri
cks
(pro
tein
gene
rato
rs a
nd i
nvert
ers
) w
hic
h w
ere
add t
o th
e R
egi
stry
, w
e h
ave t
he 9
0%
of
adva
nce o
f ou
r sy
stem
com
plete
d a
s fo
llow
s.
We a
lso
hav
e p
reelim
inay
dat
a co
nfir
min
g th
e p
ropp
er
func
tion
ing
of t
he a
ctiv
ator
mod
ule, as
show
in
the
pict
ures.
No IPTG
160 mM IPTG
Con
clus
ions
and
fur
ther
wor
kT
he p
relim
inar
y re
sult
s sh
ow t
hat
the a
ctiv
ator
sys
tem
wor
ks. T
he inh
ibit
or w
ill be c
oupl
ed b
y th
e e
nd o
f th
e y
ear
. In
order
to p
erf
orm
the e
xpe
rim
ent
s to
show
if
our
bio
bri
cks
syst
em
is
able
to
prod
uce a
n sp
a-ti
otem
pora
l pa
ttern
, w
e w
ill us
e a
chem
ical
gra
die
nt o
f IP
TG
and
aT
c. T
he m
odeling
resu
lts
hav
e g
iven
us
an ins
ight
of h
ow t
he s
yste
m w
ill re
spon
d t
o pa
rtic
ular
chem
ical
ini
tial
dis
trib
utio
ns.
Besi
des
we a
lso
need t
o co
mpl
ete
the m
odellin
g pa
rt o
f th
e p
roje
ct w
ith d
iffe
rent
pers
pect
ives.
T
he u
se
of s
toch
asti
c ap
proa
ches
shou
ld a
llow
us
to int
roduc
e n
oise
in
the s
yste
m a
nd loo
k how
the e
merg
enc
e o
f bac
teri
a pa
ttern
cou
ld b
e a
ffect
ed.
We a
re n
ot s
o fa
r to
con
clud
e if
an a
ctiv
ator
-inh
ibit
or m
ech
anis
m im
plem
ent
ed w
ith b
iobri
cks
coul
d g
ene
-ra
te a
spa
tiot
em
pora
l tu
ring
lik
e p
atte
rn.
Main r
efe
renc
es
-Tur
ing
A. M
., "T
he C
hem
ical
Bas
is o
f M
orph
ogene
sis"
, Ph
ilos
ophic
al T
rans
acti
ons
of T
he R
oyal
Soc
iety
of
Lon
don
, se
ries
B, 2
37
:37
–72
, 19
52
-Gie
rer
A. an
d M
ein
har
dt
H.,
"A t
heor
y of
bio
logy
cal pa
ttern
for
mat
ion"
, K
ybern
eti
k 12
:30
-39
, 19
72
-Mur
ray
J.D
., "M
athem
atic
al B
iolo
gy I
I",
Spr
inge
r-V
erl
ag B
erl
in H
eid
elb
erg
, 3
rd e
dit
ion,
19
93
Init
ial
inst
abilit
y dam
ping
by
the s
inui
dal
ini
tial
con
dit
ion
of d
ynam
ical
sys
tem
sim
ulat
ion.
The s
yste
m e
volu
tion
lead
to
an
nonh
omog
ene
ous
stat
e b
ut a
n sp
atia
l gr
adie
nt.
€
dPAI
[]
dt
=kcPAI
GPAITRT
ma
xF
1−TdPAI
PAI
[]−
DPAIPAI
[]
dAI
[]
dt
=kcAI
GAITRT
ma
xF
2−TdAI
AI
[]−
DAIAI
[]
F1
=
LasR
+PAI
[]2
kd
1
2
1+
LasR
+PAI
[]2
kd
1
2
1
1+
LuxR
+AI
[]2
kd
2
2
F2
=
LasR
+PAI
[]2
kd
1
2
1+
LasR
+PAI
[]2
kd
1
2
