Theoretical Biology 2016 Transcription factors bind DNA to block - - PowerPoint PPT Presentation

Theoretical Biology 2016

Chapter 7

Gene regulation

Transcription factors

bind DNA to block or enhance transcription

From Campbell

DNA makes RNA makes protein

dM dt = c − dM and dP dt = lM − δP

Golding et al. Cell 2005

Golding et al. Cell 2005

mRNA formation occurs in bursts

Number of mRNA transcripts in individual cells over time. Red: data, blue: daughter cells, drops: cell division.

P M

c d

Mathematical model: mRNA & protein

dM dt = c − dM and dP dt = lM − δP Protein Messenger

Now with negative feedback on transcription

P M

c d

Messenger Protein

dM dt = c 1 + P/h − dM and dP dt = lM − δP ,

Quasi steady state assumption

Suppose turnover of protein much faster than that of mRNA

dP dt = lM − δP = 0

• r

P = l δ M

Substituting this into dM/dt gives:

dM dt = c 1 + (l/δ)M/h − dM = c 1 + M/h0 − dM

with h’ = hδ/l

dM dt = c 1 + P/h − dM and dP dt = lM − δP ,

Cross linking of receptors activates cells

allergen

Mast cell degranulation B cells activate and start to divide

Bivalent ligand binding a monovalent receptor

C: free ligand (C > NRT), R: free receptors, RT: total receptors, C1: single bound ligand, C2: double bound ligand: RT = R + C1 + 2C2 How does C2, and hence the growth rate, depend on C? C C2

dN dt = bN 2C2 RT − dN .

?

C C2 C1 R

R C C1 C2

RT = R + C1 + 2C2 , dC1 dt = 2konRC − koffC1 − xonRC1 + 2xoffC2 , dC2 dt = xonRC1 − 2xoffC2 . To study the steady state we set dC2/dt = 0 and add this to dC1/dt: dC1 dt = 0 = 2konRC − koffC1 = 2KRC − C1 , where K = kon/koff and R = RT − C1 − 2C2. Solving this gives C1 = 2CK(RT − 2C2) 1 + 2CK ,

10 0.3 0.15 C2 100 10 0.1 0.01 0.001 0.0001

• 05

1

C1 = 2CK(RT − 2C2) 1 + 2CK , which can be substituted into dC2/dt = 0 to solve C2 as a function of C: C2 = 1 + 4CK + 4C2K2 + 4CKRTX − (1 + 2CK)

q

(1 + 2CK)2 + 8CKRTX 8CKX where X = xon/xoff.

C2 C

Thus, the number of crosslinks is a bell- shaped function of the ligand concentration C. Cells grow best at intermediate ligand concentrations

dN dt = bN 2C2 RT − dN .

dt − − dC2 dt = xonRC1 − 2xoffC2 .

Lac operon, Jacob & Monod (1961)

From Campbell

Translate this into simple scheme

• peron: 0/1

repressor allolactose permease Β-galactos. Extra- cellular Lactose Cell membrane

Towards a phenomenological mathematical model

Repressor is modeled as a declining sigmoid Hill function. We will even scale the allolactose concentration such that h=1

allolactose: A Repressor: R

I h

R = h5 h5 + A5 = 1 1 + (A/h)5

repressor allolactose

Complete mathematical model

R: repressor, M: messenger & A: allolactose:

c0: basal transcription rate, c0+c: transcription rate when operon is “on”, d and δ are decay rates of mRNA and allolactose, ML is the permease mediated influx

• vMA term: B-galactosidase hydrolizes allolactose.

R=0: operon “on” R=1: operon “off”

R = 1 1 + An , dM dt = c0 + c(1 − R) − dM = c0 + cAn 1 + An − dM , dA dt = ML − δA − vMA ,

Nullclines

A M

c0 d c0+c d

M = c0 d + (c/d)An 1 + An

M = δA L − vA

A’=0: M’=0:

sigmoid Hill function Origin: M = (δ/L)A

Quasi steady state dM/dt = 0

L ¯ A

For one concentration of L three concentrations of A

Lmin Lmax

dt − − dA dt = ML − δA − vMA

M = c0 d + (c/d)An 1 + An

with

Observed bi-stability in E. coli

Green: E. coli with high expression of lac operon From: Ozbudak et al. Nature, 2004 (see the reader)

Bi-stability in growth of E. coli

The Innate Growth Bistability and Fitness Landscapes of Antibiotic-Resistant Bacteria

• J. Barrett Deris,1,2* Minsu Kim,1*† Zhongge Zhang,3 Hiroyuki Okano,1 Rutger Hermsen,1,2‡

Alexander Groisman,1 Terence Hwa1,2,3§

Science 2013

bi-stability in algal densities in lakes

a

Fold 2 Fold 1 Phosphorus input rate,

• P concentration

Oligotrophic state Eutrophic state 0.2 0.3 0.4 0.5 0.6 0.7 0.5 1 1.5 2 Bistability region

LETTER

doi:10.1038/nature11655

Flickering gives early warning signals of a critical transition to a eutrophic lake state

Rong Wang1,2, John A. Dearing1, Peter G. Langdon1, Enlou Zhang2, Xiangdong Yang2, Vasilis Dakos3,4 & Marten Scheffer3

Nature 2012

Initiation and termination of epileptic seizures

PNAS: 2012

A i C

Post-ictal critical transition

i c t a l

s e p a r a t r i x

increasing connectivity bistability

100ms 10mV

ii iii

−2

Temporal-Corr

**

Slope

10

−2

[ ]

Mid-seizure Pre-termination

MODEL

D

1.0

Proportion

0.2 0.0 Ictal Post- Pre- Ictal Post- Pre-

** ** ** **

Spatial-Corr

*

p<0.005

** * p<0.05

1.2 0.2 3400 4200

s t a b l e l . c . stable f.p. unstable l.c.

C

Post-ictal Ictal critical transition

he i c t a l

unstable f.p.

B

Frequency

Human seizures self-terminate across spatial scales via a critical transition

Mark A. Kramera,1, Wilson Truccolob,c,d,e, Uri T. Edena, Kyle Q. Lepagea, Leigh R. Hochbergc,d,e,f,g, Emad N. Eskandarf,h, Joseph R. Madseni,j, Jong W. Leek, Atul Maheshwarid,f, Eric Halgrenl, Catherine J. Chud,f, and Sydney S. Cashd,f

Initiation and termination of depression

PNAS: 2014

autocorrelation variance autocorrelation variance correlation

A C

D

E

G

F

H

I

K

J

L

B

within-valence between-valence correlation within-valence between-valence

200 8 time emotion strength 200 8 time emotion strength

x1, x2 x3, x4 x1, x2 x3, x4

x1 x2 x3 x4 0.6 SD 0.5 1.5 0.5 1.5 x1(t) / x1 x1(t+1) / x1 x1 x2 x3 x4 0.6 SD 0.5 1.5 0.5 1.5 x1(t) / x1 x1(t+1) / x1 0.5 1.5 0.5 1.5 x1(t) / x1 x2(t) / x2 0.5 1.5 0.5 1.5 x1(t) / x1 x3(t) / x3 0.5 1.5 0.5 1.5 x1(t) / x1 x2(t) / x2 0.5 1.5 0.5 1.5 x1(t) / x1 x3(t) / x3 AR(1)=0.38 AR(1)=0.77 =0.29 =-0.47 =0.69 =-0.83

Critical slowing down as early warning for the onset and termination of depression

Ingrid A. van de Leemputa,1,2, Marieke Wichersb,1, Angélique O. J. Cramerc, Denny Borsboomc, Francis Tuerlinckxd, Peter Kuppensd,e, Egbert H. van Nesa, Wolfgang Viechtbauerb, Erik J. Giltayf, Steven H. Aggeng, Catherine Deromh,i, Nele Jacobsb,j, Kenneth S. Kendlerg,k, Han L. J. van der Maasc, Michael C. Nealeg, Frenk Peetersb, Evert Thieryl, Peter Zacharm, and Marten Scheffera

2001

• Fraction of lake surface covered

by charophyte vegetation Total P (mg l –1) 0.1 0.2 0.3 0.4 0.5 0.05 0.10 0.15 0.20 0.25 0.30

Ecosystem state Conditions Perturbation F1 F2