On the Repressilator Josef Hofbauer University College London

with Stefan M¨uller, Peter Schuster, . . . Theoretical Biochemistry Group, University Vienna The Repressilator simple : loop of three genes whose prod- ucts repress the of the following gene

A synthetic oscillatory network of transcriptional regulators Michael B. Elowitz and Stanislas Leibler, Nature (2000) design and construct a synthetic network to implement a partic- ular function use 3 transcriptional systems (not part of any natural biological clock) to build an oscillating network in . The network periodically induces the synthesis of green fluorescent protein as a readout of its state in individual cells. The resulting oscillations, with typical periods of hours, are slower than the cell-division cycle, so the state of the oscilla- tor has to be transmitted from generation to generation.

When combined with a green fluorescent reporter gene, the Re- pressilator causes growing E. coli cells to flash periodically, or twinkle, demonstrating that oscillations can be genetically pro- grammed.

first repressor protein, LacI from E. coli (lac operon, Jacob & Monod 1960) inhibits the transcription of the second repressor gene tetR from the tetracycline-resistance transposon Tn 10, whose protein product in turn inhibits the expression of a third gene, cI from λ phage. Finally, CI inhibits lacI expression, com- pleting the cycle. The repressilator: M.B. Ellowitz, S. Leibler. A synthetic oscillatory network of transcriptional regulators. Nature 403:335-338, 2002 mathematical model α m˙ = −m + + α i i h 0 1 + pi−1

p˙i = β(mi − pi)

pi repressor–protein concentrations mi mRNA concentrations (i = lacI, tetR, cI) two types of behaviour: 1) convergence to a stable steady state 2) the steady state unstable, sustained limit-cycle oscillations.

(mainly numerically) m˙ i = fi(pi−1) − δmi p˙i = β(mi − pi) fi monotone decreasing

Goodwin (1965), Fraser and Tiwari (1974): analog computer Tyson & Othmer (1978), Banks & Mahaffy (JTB 1978) Hal Smith ( JMB 1987)

Mallet-Paret & Hal Smith (1990): Poincar´e–Bendixson theorem for monotone cyclic systems Every orbit converges either to the equilibrium E or to a periodic orbit. If E is unstable then there is an asympt. stable per. orbit. open problems: If E is stable then it is globally asympt. stable. Uniqueness of periodic attractor Abstraction of the chemical processes within a living cell

(a) Gene products bind to the regulatory regions of the genes and either enhance or inhibit their expression. Binding reactions are in equilibrium, i.e. binding is faster than transcription and translation. (b) Transcription and translation are operating under saturated conditions, i.e. we do not consider the availability of nucleotides and amino acids. (c) mRNAs and proteins degrade at constant rates.

TS R r˙i = k ai − d ri TL P p¯˙i = k ri − d pi TS R r˙i = k ai − d ri TL P p¯˙i = k ri − d pi

ri concentration of the mRNA transcribed from gene Gi pi, p¯i the free (total) concentration of translated protein ai = ai(p) transcriptional activity of Gi (= concentration of gene Gi being transcribed) kTS, kTL rate constants for transcription and translation dR, dP degradation rates of mRNA and proteins repressilator: a cyclic system of n genes, each gene is repressed by its predecessor in the cycle.

More precisely, the transcription of a certain gene is repressed by the product of the preceding gene.

Two mechanisms:

The repressilator with leaky transcription In this system, genes are transcribed at a low rate, if they are repressed (leaky transcription), whereas genes are transcribed at a high rate, if they are not repressed. . . . Elowitz & Leibler model The repressilator with auto-activation and mass-action kinetics

Repressed genes are not transcribed. The transcription of a certain gene is activated by its own product. So genes are transcribed only if they are not repressed and auto- activated. Repressor and activator binding is determined by mass-action kinetics. Promotor Transcribed, processed, and State I: translated into protein

basal state Activator binding site Repressor binding site RNA polymerase

Promotor Transcribed, processed, and State II: translated into protein active state Activator Repressor binding site RNA polymerase

Active states of gene regulation Promotor State III: inactive state Activator binding site Repressor RNA polymerase

Promotor State III: inactive state Activator Repressor

RNA polymerase

Inactive states of gene regulation In the repressilator with auto-activation, each gene Gi can be bound by its own product Pi and by the product Pi−1 of the A preceding gene. The gene-activator complex Ci and the gene- R repressor complex Ci are formed: A Ci Gi + Pi (1) R Ci Gi + Pi−1 (2) transcriptional activity = concentration of the gene-activator A complex: ai = ci mass-action kinetics, binding equilibrium g p g p cA = i i cR = i i−1 i KA i KR KA,KR : dissociation constants for auto-activator and repres- sor binding I. Activator and repressor share the same binding site

A R Ci Gi + Pi,Ci Gi + Pi−1 mass conservation for genes and proteins p p ¯g = g + cA + cR = g (1 + i + i−1) (3) i i i i KA KR A R p¯i = pi + ci + ci+1 (4) g p p ¯g p p ⇒ cA = i i = i /(1 + i + i−1) (5) i KA KA KA KR relating total and free protein concentrations:  ¯g p p ¯g p p  p¯ = p 1 + /(1 + i + i−1) + /(1 + i+1 + i ) (6) i i KA KA KR KR KA KR II. Activator and repressor have different binding sites

A R Ci Gi + Pi,Ci Gi + Pi−1 AR Consider gene complex Ci containing both regulators: AR R Ci Ci + Pi (7) AR A Ci Ci + Pi−1 (8) additional relations: cAp cR p cAR = i i−1 = i i i KAR KRA

KAKAR = KRKRA (9) mass conservation for genes and proteins: p p p p ¯g = g + cA + cR + cAR = g (1 + i + i−1 + i i−1 ) (10) i i i i i KA KR KAKAR A AR R AR p¯i = pi + ci + ci + ci+1 + ci+1 (11)

g p p ¯g p p p p ⇒ cA = i i = i (1 + i + i−1 + i i−1 )−1 (12) i KA KA KA KR KAKAR relating total and free protein concentrations:  ¯g p p p p p p¯ = p 1 + (1 + i−1 )(1 + i + i−1 + i i−1 )−1 (13) i i KA KAR KA KR KAKAR ¯g p p p p p  + (1 + i+1 )(1 + i+1 + i + i+1 i )−1 KR KRA KA KR KAKAR A R special case: two independent binding sites: K = K = 1 KRA KAR  ¯g p ¯g p  p¯ = p 1 + (1 + i )−1 + (1 + i )−1 i i KA KA KR KR limit case: one binding site KA KR = → 0 (14) KRA KAR g p p ¯g p p ⇒ cA = i i = i (1 + i + i−1)−1 (15) i KA KA KA KR Rescaling: t kP kTL kTS τ = , β = , σ = (16) 1/kR kR kP kR

p r kTL x = , y = (17) KA KA kP binding ratio ρ, cooperativity parameter κ, binding parameter γ, cumulative parameter α: KA KA KR ¯g ρ = , κ = = , γ = , α = γ σ KR KRA KAR KA Repressilator with auto-activation

x¯˙i = β (yi − xi) protein conc. y˙i = α f(xi, xi−1) − yi mRNA conc.

xi f(xi, xi−1) = 1 + xi + ρ xi−1 + κ ρ xi xi−1 " !# 1 + κ ρ xi−1 ρ (1 + κ xi+1) x¯i = xi 1 + γ + 1 + xi + ρxi−1 + κρxixi−1 1 + xi+1 + ρ xi + κρxi+1xi

κ = 1: two independent binding sites " 1 ρ !# x¯i = xi 1 + γ + 1 + xi 1 + ρ xi κ = 0: limit case of one binding site Transformation x → x¯: free and total protein concentrations

    x¯˙ β (y − x)       =   y˙ α F (x) − y

∂x¯ x¯˙ = x˙ = M(x)x ˙ ∂x

    x˙ β M(x)−1 (y − x)       =   y˙ α F (x) − y M(x) is cyclically tridiagonal:   (1 + κ ρ xi−1) (1 + ρ xi−1) 1 + γ  (1 + x + ρ x + κ ρ x x )2  i i−1 i i−1  !  ρ (1 + κ xi+1) (1 + xi+1)  + if j = i  (1 + x + ρ x + κ ρ x x )2  i+1 i i+1 i ∂x¯  i ρ (κ − 1) x M(x)ij = = γ i if j = i − 1 ∂xj  2  (1 + xi + ρ xi−1 + κ ρ xi xi−1)    ρ (κ − 1) xi γ 2 if j = i + 1  (1 + xi+1 + ρ xi + κ ρ xi+1 xi)   0 otherwise The matrix M(x) is invertible.

Proof. Mi,i > |Mi−1,i| + |Mi+1,i|. That is, M(x) is diagonally (column) dominant. Hence, M(x) is invertible. The map x 7→ x¯ is one-to-one (by a global inverse function theorem) 2

For weak regulator binding, γ  1, total and free protein con- centrations become equal,x ¯i = xi and M(x) = I.     x˙ β (y − x)       =   y˙ α F (x) − y Simplified system, occurs frequently in traditional treatments of gene regulatory networks

The simplified system and the exact system have same equilibria!

    x˙ β M(x)−1 (y − x)       =   y˙ α F (x) − y Classification depends on the number of genes n, the binding ratio ρ, the cooperativity parameter κ, the cumulative parame- ter α, the degradation ratio β, and the binding parameter γ. The critical parameters are α, ρ, and κ.

α < 1: origin O is the global attractor

α > 1: interior equilibrium Ec with xi = yi = xc > 0, 2 where α = 1 + xc + ρ xc + κ ρ xc . Additionally, there are equilibria on the boundary.

ρ < 1: Ec is asymptotically stable (globally??) There are 2n − 1 boundary equilibria, all unstable.

ρ > 1: there are Ln boundary equilibria (Ln ... Lucas numbers) There is a crucial difference whether n is even or odd. n even: two stable boundary equilibria, Eodd and Eeven. All other equilibria, including Ec, are unstable. For the simplified system (M = I), almost all orbits converge to either Eodd or Eeven. (follows from Hirsch & Smith’s theory of monotone flows) n odd: all boundary equilibria are unstable. Possible attractors are (a) the central equilibrium Ec, (b) a stable limit cycle in the interior (c) a heteroclinic cycle connecting unstable equilibria in the boundary.

The transition from (a) to (b) occurs via a Hopf bifurcation. In the transition from (b) to (c), the limit cycle approaches the boundary and its period grows to infinity. Repressilator with auto-activation

(for the simplified system γ = 0 with weak regulator binding, so that x =x ¯)

x˙i = β (yi − xi) αxi y˙i = − yi 1 + xi + ρ xi−1 + κ ρ xi xi−1

β → ∞ : singular limit −→ x = y " α # x˙i = xi − 1 1 + xi + ρxi−1 + κ ρ xi xi−1 monotone cyclic feedback system −→ Mallet-Paret & Smith’s Poincar´e–Bendixson result applies again

Problem: for finite β, i.e., the full system ?? The Heteroclinic Cycle

" α # x˙i = xi − 1 1 + xi + ρxi−1 + κ ρ xi xi−1 n = 3, α > 1 and ρ > 1: similar to May–Leonard system at E1 = (α − 1, 0, 0): x˙ α α 2 = − 1 = − 1 =: −µ < 0 x2 1 + x2 + ρx1 + κρx1x2 1 + ρ(α − 1) x˙ α 3 = − 1 = α − 1 =: λ > 0 x3 1 + x3 + ρ x2 + κ ρ x2 x3

System is permanent if µ < λ Heteroclinic cycle is attracting if µ > λ . For repressilator with n odd, α > 1 and ρ > 1: n−1 (i) The system is permanent if λ > 2 µ. n−1 (ii) There is an attracting heteroclinic cycle if λ < 2 µ where: s 1 + β 1 − β2 λ = − + + β α (18) 2 2 v u 2 1 + β u1 − β α −µ = − + t + β (19) 2 2 1 + ρ (α − 1)

This heteroclinic cycle connects the following equilibria. n = 3: E1 → E3 → E2 → E1 E1 : x1 = y1 > 0, x2 = y2 = x3 = y3 = 0, . . . Proteins mRNAs

0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0 1e+07 2e+07 3e+07 4e+07 5e+07 0 1e+07 2e+07 3e+07 4e+07 5e+07 0.3 0.08 0.25

0.06 0.2

0.15 0.04 0.1 0.02 0.05

0 0 0 1e+07 2e+07 3e+07 4e+07 5e+07 0 1e+07 2e+07 3e+07 4e+07 5e+07

The repressilator limit cycle Proteins mRNAs

1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0 1 100 10000 1e+06 1e+08 1 100 10000 1e+06 1e+08 0.3 0.3

0.25 0.25 The repressilator heteroclinic cycle for = 3 (logarithmic timescale) 0.2 0.2 n

0.15 0.15

0.1 0.1

0.05 0.05

0 0 1 100 10000 1e+06 1e+08 1 100 10000 1e+06 1e+08

The repressilator heteroclinic orbit (logarithmic time scale) This heteroclinic cycle connects the following equilibria. n = 3: E1 → E3 → E2 → E1 n = 5: E13 → E35 → E52 → E24 → E41 → E13 n = 7: E135 → E357 → E572 → E724 → E246 → E461 → E613 → E135

In general, the heteroclinic cycle connects the n equilibria Ei,i+2,i+4,...,i+n−3, whose support has the pattern ∗0 ∗ 0 · · · ∗ 00 (cyclically modulo n). All orbits on the boundary converge to an equilibrium.

Proof: Suppose xi−1 = yi−1 = 0, xi + yi > 0, then xi(t) → α − 1 and yi(t) → α − 1. xi(t) close to α−1, repression leads to removal of next gene/protein: y˙i+1 < xi+1 − yi+1 yields: · (xi+1 + β yi+1) < 0 (20)

Hence, xi+1(t) → 0 and yi+1(t) → 0. ⇒ convergence to a boundary equilibrium. 2

If only one species (say i − 1) is missing initially, then the limit equilibrium has support {i, i + 2, . . . , i + n − 3}, and is a member of the heteroclinic cycle described above. Let S be the support of a boundary equilibrium ES, and S + 1 the set of all successors of elements in S. At ES, each i ∈ S+1 is repressed and hence its decline is governed by the linearized system α x˙ = β (y − x ), y˙ = x − y i i i i 1 + ρ (α − 1) i i with leading eigenvalue −µ < 0. Let (1, cµ) (with cµ > 0) be the corresponding left eigenvector, so that · (xi + cµ yi) = −µ (xi + cµ yi) (21) holds near ES.

If neither i nor i − 1 are in S, then i is not repressed, and hence its invasion is governed by the linearized system

x˙i = β (yi − xi), y˙i = α xi − yi If neither i nor i − 1 are in S, then i is not repressed, and hence its invasion is governed by the linearized system

x˙i = β (yi − xi), y˙i = α xi − yi whose leading eigenvalue λ is positive. Let (1, cλ) (with cλ > 0) be the corresponding left eigenvector, so that · (xi + cλ yi) = λ (xi + cλ yi) (22) holds near ES. Let z = (x1, y1, . . . , xn, yn). We use the function n Y P (z) = (xi + ci(z) yi) (23) i=1 as an average Lyapunov function.

Choose ci(·) in a smooth way, s.t. ci(z) = cµ for z close to any boundary equilibrium ES with i − 1 ∈ S and ci(z) = cλ for z close to any boundary equilibrium ES with i − 1 ∈/ S.

˙ P (z) for all and (21) and (22) imply that near P (z) ≥ −m z ES n · P˙ (z) X (xi + ci(z) yi) = = λ (n−2 |S|)−µ |S|+O(|z−ES|) (24) P (z) i=1 (xi + ci(z) yi) n · P˙ (z) X (xi + ci(z) yi) = = λ (n−2 |S|)−µ |S|+O(|z−ES|) (25) P (z) i=1 (xi + ci(z) yi) n−1 n odd implies |S| ≤ 2 , so the coefficient of λ is at least 1. ˙ Hence, if n−1 then P (z) 0 near all boundary equilibria, λ > 2 µ P (z) > and the system is permanent.

˙ On the other hand, if n−1 then P (z) 0 near the equi- λ < 2 µ P (z) < n−1 libria ES with |S| = 2 , i.e., those in the heteroclinic cycle described before. Since this heteroclinic cycle is asymptotically stable within the maximal invariant set of the boundary of the 2n state space R+ , it is asymptotically stable for the full system. Summary: Repressilator with auto-activation

We derived the model from basic principles of chemical reaction kinetics.

For n odd, α, ρ > 1 we can locate 4 regimes in parameter space: a) Ec stable, h.c. repelling globally asympt. stable ?? b) Ec unstable, system permanent −→ periodic attractor ?? c) Ec unstable, attracting het. cycle globally attracting ?? d) Ec stable, het. cycle attracting Ρ n3, Κ10 10

8

6

4

Α 1.2 1.4 1.6 1.8 2 bifurcation diagram for n = 3, κ = 10, β → ∞: Ec is stable below the red line the heteroclinic cycle is attractive above the green curve