Design Principles of Genetic Circuits Justin Bois Caltech Spring, 2018

BE 150: Design Principles of Genetic Circuits Justin Bois Caltech Spring, 2018

This document was prepared at Caltech with ﬁnancial support from the Donna and Benjamin M. Rosen Bioengineering Center.

Today we will talk about an oscillating genetic circuit developed by Elowitz and Leibler and published in 2000, called the repressilator. As we work through this example, we will learn a valuable technique for analyzing dynamical systems, including many of those we encounter in systems biology, linear stability analysis.

8.1 Design of the repressilator

The repressilator consists of three repressors on a plasmid, as shown in Fig. 7. They are TetR, λ cI, and LacI. For our analysis here, the names are unimportant, and we will call them repressor 1, 2, and 3. Repressorletters 1 represses to production nature of repressor 2, which in turn represses production of repressor 3. Finally, repressor 3 represses production or repressor 1, completing the loop.

a Repressilator Reporter 30 8C. At least 100 individual cell lineages in each of three micro- colonies were tracked manually, and their ¯uorescence intensity was P lac01 L quanti®ed. ampR tetR-lite The timecourse of the ¯uorescence of one such cell is shown in PLtet01 Fig. 2. Temporal oscillations (in this case superimposed on an kanR overall increase in ¯uorescence) occur with a period of around Tet R Tet R pSC101 gfp-aav 150 minutes, roughly threefold longer than the typical cell-division λP origin R time. The amplitude of oscillations is large compared with baseline λ cI LacI GFP levels of GFP ¯uorescence. At least 40% of cells were found to exhibit oscillatory behaviour in each of the three movies, as lacI-lite ColE1 determined by a Fourier analysis criterion (see Methods). The λ cI-lite range of periods, as estimated by the distribution of peak-to-peak intervals, is 160 6 40 min (mean 6 s:d:, n 63). After septation, PLtet01 GFP levels in the two sibling cells often remained correlated with Figure 7: Schmematic of the plasmids used to construct the repessilator in E. coli. The one another for long periods of time (Fig. 3a±c). Based on the three repressors TetR, λ cI, and LacI, are on the same plasmid and form a cycle of repression. Additionally, TetR represses GFP, which is found on another plasmid. The analysis of 179 septation events in the 3 movies, we measured an lite sufx on the repressors signiﬁes that they have a destruction tage to decrease their average half-time for sibling decorrelation of 95 6 10 min, which is stability.b The aav sufx on the GFP indicates that it is a variant of intermediate stability. longer than the typical cell-division times of 50±70 min under these Taken from Elowitz and Leibler, Science, 403, 335–338, 2000.

β conditions. This indicates that the state of the network is trans- steady state A We might work out thestable dynamics of this system by reasoning. We might get a stable steady state, mitted to the progeny cells, despite a strong noise component. We where all three levels of repressor are tuned to reasonably repress the others. Conversely, we might observed signi®cant variations in the period and amplitude of the image a dynamic scenario. Say that initially repressor 1 has high copy number and repressors 2 and oscillator output both from cell to cell (Fig. 3d), and over time in a 3 are low. The high copy of number of repressor 1 will keep theB numbers of repressor 2 down. This means that repressor 3 is free to be expressed. As its copy number grows, it will start to repress single cell and its descendants (Fig. 3a±c). In some individuals, repressor 1. As repressor 1 goes down, repressor 2 is expressed in higher numbers. The increased periods were omitted or phase delayed in one cell relative to its repressor 2 copy number leads to less repressor 3. Then, repressor 1 comes back up again. So, we see sibling (Fig. 3a, c). a cycle, where repressor 1 is high, then repressor 3, and ﬁnally repressor 2. Since repressor 1 represses Recent theoretical work has shown that stochastic effects may GFP, we will see an oscillation in GFP as repressor 1 goes up and down. C be responsible for noisy operation in natural gene-expression networks12. Simulations of the repressilator that take into account

Protein lifetime/mRNA lifetime, steady state 19 unstable the stochastic nature of reaction events and discreteness of network components also exhibit signi®cant variability, reducing the corre- lation time for oscillations from in®nity (in the continuous model)

Maximum proteins per cell, α (× KM) to about two periods (Box 1, Fig. 1c, insets). In general, we would like to distinguish such stochastic effects from possible intrinsically complex dynamics (such as intermittence or chaotic behaviour). Further studies are needed to identify and characterize the sources of ¯uctuations in the repressilator and other designed networks. In c particular, longer experiments performed under chemostatic con- 6,000 6,000 1 1 ditions should enable more complete statistical characterization of

0 0 4,000 4,000 -10 500 1,000 -10 500 1,000 Time (min) 2,000 2,000 60 140 250 300 390 450 550 600 a Proteins perProteins cell 0 0 0 500 1,000 0 500 1000 Time (min) Time (min) b

Construction, design and simulation of the repressilator. , The repressilator Figure 1 a 120 network. The repressilator is a cyclic negative-feedback loop composed of three repressor c genes and their corresponding promoters, as shown schematically in the centre of the 100 left-hand plasmid. It uses PLlacO1 and PLtetO1, which are strong, tightly repressible 80 6 promoters containing lac and tet operators, respectively , as well as PR, the right promoter 60 from phage l (see Methods). The stability of the three repressors is reduced by the presence of destruction tags (denoted `lite'). The compatible reporter plasmid (right) 40 11 Fluorescence expresses an intermediate-stability GFP variant (gfp-aav). In both plasmids, transcrip- units) (arbitrary 20 tional units are isolated from neighbouring regions by T1 terminators from the E. coli rrnB 0 operon (black boxes). b, Stability diagram for a continuous symmetric repressilator model 0 100 200 300 400 500 600 (Box 1). The parameter space is divided into two regions in which the steady state is stable Time (min) (top left) or unstable (bottom right). Curves A, B and C mark the boundaries between the

two regions for different parameter values: A, n 2:1, a0 0; B, n 2, a0 0; C, Figure 2 Repressilation in living bacteria. a, b, The growth and timecourse of GFP 2 3 n 2, a0=a 10 . The unstable region (A), which includes unstable regions (B) and expression for a single cell of E. coli host strain MC4100 containing the repressilator (C), is shaded. c, Oscillations in the levels of the three repressor proteins, as obtained by plasmids (Fig. 1a). Snapshots of a growing microcolony were taken periodically both in numerical integration. Left, a set of typical parameter values, marked by the `X' in b, were ¯uorescence (a) and bright-®eld (b). c, The pictures in a and b correspond to peaks and used to solve the continuous model. Right, a similar set of parameters was used to solve a troughs in the timecourse of GFP ¯uorescence density of the selected cell. Scale bar, stochastic version of the model (Box 1). Colour coding is as in a. Insets show the 4 mm. Bars at the bottom of c indicate the timing of septation events, as estimated from normalized autocorrelation function of the ®rst repressor species. bright-®eld images.

336 NATURE | VOL 403 | 20 JANUARY 2000 | www.nature.com 8.2 Dynamical equations for the repressilator

To analyze the repressilator, we will write down our usual dynamical expressions. For simplicity, we will assume symmetry among the species and will consider only protein dynamics, ignoring mRNA.

dx1 = n x1, (8.1) dt 1 +(x3/k) −

dx2 = n x2, (8.2) dt 1 +(x1/k) −

dx3 = n x3. (8.3) dt 1 +(x2/k) − In dimensionless units, this is

dxi = n xi, with i, j pairs (1, 3), (2, 1), (3, 2). (8.4) dt 1 + xj −

8.2.1 Fixed point

To ﬁnd the ﬁxed point of the repressilator, we solve for x with x˙ = 0 i. We get that i i ∀ x1 = n , (8.5) 1 + x3 x2 = n , (8.6) 1 + x1 x3 = n . (8.7) 1 + x2

We can substitute the expression for x3 into that for x1 to get x1 = . (8.8) n 1 + 1 + xn ! 2 " We can then substitute the expression for for x2 to get x1 = n . (8.9) 1 + ⎛ ⎞ n 1 + ⎜ n ⎟ ⎜ 1 + x1 ⎟ ⎝ ! " ⎠ This looks like a gnarly expression, but we can write it conveniently as a composite function. Specif- ically,

x = f(f(f(x ))) fff(x ), (8.10) 1 1 ≡ 1 where f(x)= . (8.11) 1 + xn

20 By symmetry, this relation holds for repressors 2 and 3 as well, so we have

xi = fff(xi). (8.12)

Writing the relationship for the ﬁxed point with a composite function is useful because we can easily compute the derivatives of the composite function using the chain rule.

ff ′(x)=f ′(f(x)) f ′(x), (8.13) ·

fff ′(x)=f ′(ff(x)) ff ′(x)=f ′(f(f(x))) f ′(f(x)) f ′(x). (8.14) · · ·

Now, since f(x) is monotonically decreasing, f ′(x) < 0, and also f ′(f(x)) < 0. This means that ff ′(x) > 0, so ff(x) is monotonically increasing. Now, f ′(ff(x)) < 0, since f ′(anything monotonically increasing) < 0. this means that fff(x) is monotonically decreasing. Since xi is increasing, there is a single ﬁxed point with x = fff(x) (see Fig. 8). So, we have a unique ﬁxed point with x1 = x2 = x3 x0 = n , (8.15) ≡ 1 + x0 or

n = x0(1 + x0). (8.16)

Figure 8: Sketch of the ﬁxed point of a repressilator.

Because we have a single ﬁxed point, we cannot have bistability in the repressilator. What happens at this ﬁxed point? To answer this question, we turn to linear stability analysis.

8.3 Linear stability analysis

We first give an introduction to the technique of linear stability analysis generically. The basic idea is that we approximate a nonlinear dynamical system by its Taylor series to first order near the fixed point and then look at the behavior of the simpler linear system. The Hartman-Grobman theorem (which we will not derive here) ensures that the linearized system faithfully represents the phase portrait of the full nonlinear system near the fixed point. Say we have a dynamical system with variables u with

du = f(u), (8.17) dt

21 where f(u) is a vector-valued function, i.e.,

f(u)=(f 1(u1, u2,...), f 2(u1, u2,...),...). (8.18)

Say that we have a ﬁxed point u0. Then, linear stability analysis proceeds with the following steps.

1) Linearize about u0, deﬁning u = u u0. To do this, expand f(u) in a Taylor series about u0 to ﬁrst order. −

f(u)=f(u )+ f(u ) u + , (8.19) 0 ∇ 0 · ··· where f(u ) A is the Jacobi matrix, ∇ 0 ≡

∂f1 ∂f1 ∂u1 ∂u2 ··· ∂f2 ∂f2 f(u0) A = ⎛ ⎞ . (8.20) ∇ ≡ ∂u1 ∂u2 ··· . . . ⎜ . . ..⎟ ⎜ ⎟ ⎝ ⎠ Thus, we have du du du = 0 + = f(u )+A u + higher order terms. (8.21) dt dt dt 0 · Since du 0 = f(u )=0, (8.22) dt 0 we have, to linear order, du = A u. (8.23) dt ·

2) Compute the eigenvalues, of A.

3) – If Re() < 0 for all , then the ﬁxed point u0 is linearly stable.

– If Re() > 0 for any , then the ﬁxed point u0 is linearly unstable. – If Re()=0 for one or more , with the rest having Re() < 0, then the ﬁxed point u0 lies at a bifurcation.

So, if we can assess the dynamics of the linearized system near the ﬁxed point, we can get an idea what is happening with the full system. To do the linearization, we need to do Taylor expansions of Hill functions. We do this so often, this is worth stating here and memorizing for future use.

n n 1 x nx0− n = x0 + n 2 x + higher order terms, (8.24) 1 + x (1 + x0)

n 1 1 nx0− n = x0 n 2 x + higher order terms. (8.25) 1 + x − (1 + x0)

22 8.4 Linear stability analysis for the repressilator

To perform linear stability analysis for the repressilator, we begin by writing the linearized system.

n 1 dx1 nx0− n 2 x3 x1, (8.26) dt ≈−(1 + x0) −

n 1 dx2 nx0− n 2 x1 x2, (8.27) dt ≈−(1 + x0) −

n 1 dx3 nx0− n 2 x2 x3. (8.28) dt ≈−(1 + x0) − Deﬁning

n 1 nx0− a = n 2 , (8.29) (1 + x0) we can write this in matrix form as x x d 1 1 x = A x , (8.30) dt ⎛ 2⎞ − · ⎛ 2⎞ x3 x3 ⎝ ⎠ ⎝ ⎠ with 10a A = a 10. (8.31) ⎛ ⎞ − 0 a 1 ⎝ ⎠ To compute the eigenvalues of A, we compute the characteristic polynomial using cofactors,

(1 + )(1 + )2 + a(a2)=(1 + )3 + a3 = 0. (8.32)

This is solved to give

= 1 + a√3 1. (8.33) − − Recalling that there are three cube roots of 1, we get our three eigenvalues. − a a = 1 a, 1 + (1 + i√3), 1 + (1 i√3). (8.34) − − − 2 − 2 − The ﬁrst eigenvalue is always real and negative. The second two have a positive real part if a > 2, or

n 1 nx0− n 2 > 2. (8.35) (1 + x0) Now,

n = x0(1 + x0), (8.36) which we found when computing the ﬁxed point, so

n nx0 a = n . (8.37) 1 + x0

23 So, a > 2 only if n > 2, meaning that we must have ultrasensitivity for the ﬁxed point to be unstable. At the bifurcation,

n nx0 a = n = 2, (8.38) 1 + x0 so 2 xn = . (8.39) 0 n 2 − n Using = x0(1 + x0), we can write

n+1 n n n = 1 − (8.40) 2 2 − ) * at the bifurcation. So, for n > 2 and

n+1 n n n > 1 − , (8.41) 2 2 − ) * we have imaginary eigenvalues with positive real parts. This is an oscillatory instability.

8.5 Numerical solution of the repressilator dynamics

We can solve for the dynamics of the repressilator numerically. This is done in the Jupyter notebook accompanying this lecture. Importantly, we see that we get a succession of peaks, 1 2 3 1 2 3 . This is like a clock. Can we have a clock with 12 peaks (“hours”)? → → → → → ···