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

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LasR

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[]2

kd

1

2

1+

LasR

+PAI

[]2

kd

1

2