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

Chapter 7 Gene regulation Theoretical Biology 2016

Transcription factors bind DNA to block or enhance transcription From Campbell

DNA makes RNA makes protein Golding et al. Cell 2005 d M d P d t = lM − δ P d t = c − dM and

mRNA formation occurs in bursts Golding et al. Cell 2005 Number of mRNA transcripts in individual cells over time. Red: data, blue: daughter cells, drops: cell division.

Mathematical model: mRNA & protein Messenger M c d Protein P d M d P d t = lM − δ P d t = c − dM and

Now with negative feedback on transcription c d Messenger M Protein P d M c d P d t = 1 + P/h − dM and d t = lM − δ P ,

Quasi steady state assumption d M c d P d t = 1 + P/h − dM and d t = lM − δ P , Suppose turnover of protein much faster than that of mRNA d P P = l or d t = lM − δ P = 0 δ M Substituting this into d M /d t gives: d M c c d t = 1 + ( l/ δ ) M/h − dM = 1 + M/h 0 − dM with h’ = h δ /l

Cross linking of receptors activates cells Mast cell degranulation B cells activate and start to divide allergen

Bivalent ligand binding a monovalent receptor C : free ligand ( C > NR T ), d N d t = bN 2 C 2 R : free receptors, − dN . R T : total receptors, R T C 1 : single bound ligand, C 2 : double bound ligand: R T = R + C 1 + 2C 2 C How does C 2 , and hence the C 1 growth rate, depend on C ? C 2 R ? C 2 C

R T = R + C 1 + 2 C 2 , d C 1 = 2 k on RC − k o ff C 1 − x on RC 1 + 2 x o ff C 2 , d t d C 2 = x on RC 1 − 2 x o ff C 2 . d t To study the steady state we set d C 2 / d t = 0 and add this to d C 1 / d t : d C 1 d t = 0 = 2 k on RC − k o ff C 1 = 2 KRC − C 1 , C C 1 where K = k on /k o ff and R = R T − C 1 − 2 C 2 . C 2 Solving this gives R C 1 = 2 CK ( R T − 2 C 2 ) , 1 + 2 CK

− − d t C 1 = 2 CK ( R T − 2 C 2 ) d C 2 , = x on RC 1 − 2 x o ff C 2 . 1 + 2 CK d t which can be substituted into d C 2 / d t = 0 to solve C 2 as a function of C : 1 + 4 CK + 4 C 2 K 2 + 4 CKR T X − (1 + 2 CK ) q (1 + 2 CK ) 2 + 8 CKR T X C 2 = 8 CKX where X = x on /x o ff . 0.3 Thus, the number of C 2 C2 crosslinks is a bell- shaped function of the ligand concentration C. 0.15 Cells grow best at intermediate ligand concentrations d N d t = bN 2 C 2 C − dN . 0 R T -05 10 0.0001 0.001 0.01 0.1 1 10 100

Lac operon, Jacob & Monod (1961) From Campbell

Translate this into simple scheme Cell membrane operon: 0/1 Β -galactos. repressor permease Extra- allolactose cellular Lactose

Towards a I repressor phenomenological mathematical model Repressor: R allolactose h 5 1 R = h 5 + A 5 = 1 + ( A/h ) 5 0 h allolactose : A Repressor is modeled as a declining sigmoid Hill function. We will even scale the allolactose concentration such that h= 1

Complete mathematical model R : repressor, M : messenger & A : allolactose: R =0: operon “on” 1 R = 1 + A n , R =1: operon “off” cA n d M = c 0 + c (1 − R ) − dM = c 0 + 1 + A n − dM , d t d A = ML − δ A − vMA , d t c 0 : basal transcription rate, c 0 + 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.

Nullclines c 0 + c d A ’=0: M δ A M = L − vA Origin: M = ( δ /L)A c 0 d A M ’=0: d + ( c/d ) A n M = c 0 sigmoid Hill function 1 + A n

Quasi steady state d M /d t = 0 ¯ A − − d t d A = ML − δ A − vMA d t with d + ( c/d ) A n M = c 0 1 + A n L min L max L For one concentration of L three concentrations of A

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 Hwa 1,2,3 § Science 2013

bi-stability in algal densities in lakes 2 Bistability region a Eutrophic state 1.5 P concentration Fold 2 1 Fold 1 0.5 Oligotrophic state 0 0.2 0.3 0.4 0.5 0.6 0.7 LETTER Phosphorus input rate, � doi:10.1038/nature11655 Flickering gives early warning signals of a critical transition to a eutrophic lake state Nature 2012 Rong Wang 1,2 , John A. Dearing 1 , Peter G. Langdon 1 , Enlou Zhang 2 , Xiangdong Yang 2 , Vasilis Dakos 3,4 & Marten Scheffer 3 �

Initiation and termination of epileptic seizures A i B iii ii bistability ** l ** l a a t t 0 c c critical transition i i r a t r i a x p [ ] e Temporal-Corr − 2 Spatial-Corr s Frequency 10 Post-ictal Slope − 2 10mV 100ms ** increasing connectivity C D Ictal critical transition MODEL s t a b l e l . c . 1.0 1.2 Mid-seizure Pre-termination Proportion unstable l.c. p<0.005 ** Post-ictal * p<0.05 he unstable f.p. stable f.p. ** ** * 0.2 0.2 0.0 Post- Post- Ictal Ictal Pre- Pre- 3400 C 4200 Human seizures self-terminate across spatial scales PNAS: 2012 via a critical transition Mark A. Kramer a,1 , Wilson Truccolo b,c,d,e , Uri T. Eden a , Kyle Q. Lepage a , Leigh R. Hochberg c,d,e,f,g , Emad N. Eskandar f,h , Joseph R. Madsen i,j , Jong W. Lee k , Atul Maheshwari d,f , Eric Halgren l , Catherine J. Chu d,f , and Sydney S. Cash d,f

Initiation and termination of depression A B D C 8 8 emotion strength emotion strength x 1 , x 2 x 1 , x 2 x 3 , x 4 x 3 , x 4 0 0 0 200 0 200 time time G E F H variance autocorrelation variance autocorrelation 0.6 0.6 1.5 1.5 x 1 (t+1) / x 1 x 1 (t+1) / x 1 SD SD AR(1)=0.38 AR(1)=0.77 0 0 0.5 0.5 x 1 x 2 x 3 x 4 x 1 x 2 x 3 x 4 0.5 1.5 0.5 1.5 x 1 (t) / x 1 x 1 (t) / x 1 correlation correlation K I J L within-valence between-valence within-valence between-valence 1.5 1.5 1.5 1.5 x 2 (t) / x 2 x 2 (t) / x 2 x 3 (t) / x 3 x 3 (t) / x 3 � =0.29 � =-0.47 � =0.69 � =-0.83 0.5 0.5 0.5 0.5 0.5 1.5 0.5 1.5 0.5 1.5 0.5 1.5 x 1 (t) / x 1 x 1 (t) / x 1 x 1 (t) / x 1 x 1 (t) / x 1 Critical slowing down as early warning for the onset and termination of depression PNAS: 2014 Ingrid A. van de Leemput a,1,2 , Marieke Wichers b,1 , Angélique O. J. Cramer c , Denny Borsboom c , Francis Tuerlinckx d , Peter Kuppens d,e , Egbert H. van Nes a , Wolfgang Viechtbauer b , Erik J. Giltay f , Steven H. Aggen g , Catherine Derom h,i , Nele Jacobs b,j , Kenneth S. Kendler g,k , Han L. J. van der Maas c , Michael C. Neale g , Frenk Peeters b , Evert Thiery l , Peter Zachar m , and Marten Scheffer a