Problem formulation

solution. To increase the robustness of clock with respect to this molecular load, an “insulator cir- cuit” was developed to put only a small load on the oscillator, amplify the measured signal, and drive larger downstream loads. The project presented in this chapter was developed in close collaboration with J. Kim, E. Winfree and the group of Prof. Friedrich Simmel at the Technical University in Munich. The data presented here were collected by me, unless otherwise noted. Figures are adapted from [41].

computed oscillatory domain when m =n and α =β. Unless otherwise noted, from now on the operating point of this oscillator model is defined by the parameters k_p = 0.05/s, k_d = 0.002/s, KA=KI=0.5µM, [SW21^tot] = [SW12^tot] = 100 nM, m=n=5, andτ= 500 s.

We want to use one of the oscillator component species to bind to a “load” molecule L, driving the periodic formation of an “active” complex L^a. We assume that [L^tot] = [L] + [L^a]. We will make a distinction on whether the mass of the oscillator component driving the load is consumed or not by binding to the load. Such a distinction is relevant with respect to our experimental system, as we will remark later. Without loss of generality, we will consider the case where the species rI2 is coupled to the load L. Additional chemical reactions are now present in the system. The active form of the load is produced according to the second-order reaction: rI2 + L−→^kf L^a. The active load decays to its inactive form: L^{a k}−→^r L. If the mass of the oscillator species is not consumed, the previous reaction is replaced by: L^{a k}−→^r rI2 + L. In both cases, the concentration dynamics for L are:

d[L^a]

dt =−k_r·[L^a] + k_f·[L][rI2]. (3.1) The rI2 concentration dynamics are perturbed by the new reactions:

d[rI2]

dt = k_p·[SW21]−k_d·[rI2] +k_r·[L^a]

consumptive

z }| {

−k_f·[L][rI2]

| {z }

non−consumptive

, (3.2)

where the braces highlight the additional terms appearing in the consumptive and non-consumptive coupling cases.

Let us for now ignore the perturbation introduced by the presence of L on the oscillator, and assume that [rI2(t)] is unaffected by L. Then, we can approximate the solution of equation (3.1) with the quasi-steady-state expression:

[cL^a](t) = [L^tot]

1− k_r

kr+ kf[rI2(t)]

. (3.3)

The above approximation is satisfactory when the load binding rates are faster than the timescale of the oscillator, as shown in detail in the Appendix, Section 3.7.1.2. Here, for illustrative purposes, we choose kr ≈ kd = 0.006/s. Therefore, the load dynamics are always faster than the dynamics of rI2, and converge to a periodic orbit forced by the oscillatory input (this is demonstrated in the Appendix, Section 3.7.1.2.

Referring to equation (3.3), we can make some considerations on the “signal propagation” from the oscillator to the load. Suppose [rI2](t)≈A0+A1sinωt: then, the ratio of krand kf[rI2] influences

K L

rA1 SW21

rI2 SW12

L Ins ^Ins_Out

M N

0 100 200 300

0 0.5 1 1.5 2 2.5

Minutes

rI2 [µM]

No Ins, No Load Ins 50 nM, Load 0.5 µM Ins 100 nM, Load 1 µM Ins 200 nM, Load 2 µM

0 100 200 300

0 0.5 1

Minutes

La [µM] Ins 200 nM, Load 2 µM

Ins 100 nM, Load 1 µM

Ins 50 nM, Load 0.5 µM

10⁻⁶ 10⁻⁴

0 0.2 0.4 0.6 0.8 1

A₀: 0.81 to 1.3 µM

kr/k f [M]

[µM]

Mean of L^a Amplitude of L^a

I

n=m

α = β

0 2 4 6 8 10 0

2 4 6 8

10 No Ins, no Load Ins 100 nM, Load 1 µM Ins 200 nM, Load 2 µM

A

E

D B

F

rA1 SW21

rI2 SW12

rA1 SW21

rI2 SW12

L

C

G H

0 100 200 300

0 0.5 1 1.5 2 2.5

Minutes

[µM]

rA1 rI2

0 100 200 300

0 20 40 60 80 100

Minutes

[nM]

SW12 SW21

0 100 200 300

0 0.5 1 1.5 2 2.5

Minutes

rI2 [µM]

No Load Load 0.5 µM Load 1 µM Load 2 µM

0 100 200 300

0 0.2 0.4 0.6

Minutes La [µM]

Load 0.5 µM Load 1 µM Load 2 µM

n=m

α = β

0 2 4 6 8 10 0

2 4 6 8 10

Oscillating

n=m

α = β

0 2 4 6 8 10 0

2 4 6 8 10

No Load Load 1 µM Load 2 µM

RNA [µM] Switch [nM]

1 2 3 4

0 0.2 0.4 0.6 0.8 1

A₀ [µ M]

[µM]

k_r/k_f=0.05 to 1 µM

Mean of L^a Amplitude of L^a

J

A ₀ [µM]

Figure 3.1: Circuits and simulations for a simple oscillator system coupled to a load. Unless otherwise noted, the parameters used for all simulations in this panel are: k_p = 0.05/s, k_d= 0.002/s, KA=KI=0.5µM, [SW21^tot] = [SW12^tot] = 100 nM, m=n=5,τ = 500 s, kr = 0.006/s, kf = 7.9·10³/M/s. For the insulating gene, the RNA output production rate is kⁱ_p = 0.15/s, and the RNA degradation rate is kⁱ_d= 0.006/s. The consumptive binding rates of the insulator and rI2 are chosen as kr = 0.006/s and kf = 7.9·10³/M/s. The binding rates of the insulator RNA output and the load are chosen as kⁱ_r = 0.006/s and kⁱ_f = 6·10³/M/s. A. Diagram for the simple model for the oscillator. B. Time traces for the oscillator species rA1 and rI2. C.

Time traces for the oscillator species SW12 and SW21. D. Oscillatory domain of the simple model as a function of the non-dimensional parametersα=β and m = n. E. Oscillator scheme with consumptive load coupled to rI2. F and G. Time traces for the oscillator and load for consumptive coupling on rI2. H. The oscillatory domain shrinks as a function of [L^tot] for the consumptive coupling to rI2. I. Mean and amplitude of the active load [L^a] as a function of the ratio of krand kf, when the driving input is rI2 = A0+ A1sinωt, with A0varying between 0.81 (light color) and 1.3µM, and A1=.8µM,ω= 0.001 rad/s. J. Mean and amplitude of the active load signal [L^a] as a function of the baseline A0 for the input oscillating signal, for ratios kr/kf

varying between 0.05 and 1µM. K. Oscillator scheme with consumptive insulating circuit and consumptive load. L and M. Time traces for the oscillator and load when the insulating genelet is used to amplify rI2. N. The perturbation of the oscillatory domain is reduced by using a small amount of an additional genelet (insulator) that amplifies the oscillatory signal. Figure reproduced from [41]

the amplitude of the load signal as shown in Figure 3.1 I and J. In particular, we can derive the ratio k_r/k_f = p

(A²₀−A²₁), which maximizes the amplitude of [cL^a]. Therefore, for A₀ ≈ 1.1µM and A1≈0.8µM, if we choose kr= 0.006≈3·kd, then kf ≈7.9·10³ is the optimal binding rate.

However, a larger kf will increase the mean value of [L^a]. Another observation is that a high mean A0 results in a lower load amplitude (Figure 3.1 I and J).

Under the assumption that the load dynamics are well approximated with their stationary solu- tion, we can write new expressions for the perturbed dynamics of rI2. For the consumptive case we have:

d[rI2]c

dt = kp·[SW21]−kd·[rI2]c − kr·[L^tot] ^kf[rI2]c

kr+kf[rI2]c , (3.4) where the box highlights the stationary perturbation term. This term is bounded by the constant kr[L^tot], and converges to it for large values of kf[rI2]. Loosely speaking, adding the load is similarc to introducing in the rI2 dynamics a new degradation term, directly proportional to the total load amount. While the approximated trajectory (3.4) provides qualitative insight on the system behav- ior, we report the full numerical simulations of the five ordinary differential equations describing the oscillator with load in Figure 3.1, which shows the rI2 and load trajectories for increasing [L^tot].

The oscillatory domain of the system is consequently altered as shown in Figure 3.1 H. Numerically simulated time traces of the oscillator and of the load are shown in Figure 3.1 F and G.

For the non-consumptive case, it is easy to see that the sum of the approximated perturbation terms is equal to zero. Therefore, we can conclude that after a transient the dynamics of the oscillator are unaffected by the presence of the load. Numerical simulations that testify this result are reported in the Appendix, Figure 3.10.

It is important to emphasize that the non-consumptive case has been previously analyzed in [27].

By assuming rigorous time-scale separation of the load dynamics relative to the driving chemical species, the authors were able to derive a general expression for the “retroactivity” caused by the load. Although not derived under the same assumptions on the parameters, our conclusions are consistent with the results reported in that work, where the retroactivity can be minimized by choosing appropriately fast binding rates and by reducing the total load amount.

In practical cases it may be impossible to couple non-consumptively a signaling molecule to the desired load. It may also not be possible to adjust the binding rates arbitrarily to provide small retroactivity and good signal transmission. If we fall in the consumptive load coupling case with limited freedom in tuning kf and kr, expression (3.4) shows that the only way to bound the perturbation on rI2 is to reduce [L^tot]. We can overcome this limit by using rI2 to activate another genelet, whose RNA output amplifies the oscillator signal and can drive larger amounts of [L^tot].

The genelet effectively acts as an “insulator” and will be denoted as Ins. We assume that the genelet Ins binds to rI2 consumptively: rI2 + Ins−→^kf Ins^a, Ins^a −→^kr Ins. The active genelet Ins^aproduces an RNA output similarly to the oscillator switches: Ins^{a k}

i

−→p Ins^a+ InsOut. We finally assume that the RNA output, which in practice amplifies the oscillatory signal, in turn activates the desired load by the usual consumptive binding mechanism: InsOut + L ^k

i

−→f L^a, L^{a k}

i

−→r L. The RNA output is also degraded as the other RNA species in the system: InsOut−→^kid ∅. The full set of dynamic equations are reported in the Appendix, Section 3.7.1.3. For illustrative purposes we assume that kⁱ_p= 0.16/s, kⁱ_r = 0.006/s, and kⁱ_f = 6·10³/M/s. As shown in Figure 3.1 L and M, using a small amount of insulator genelet it is possible to drive large amounts of load introducing negligible perturbations.

The oscillatory domain of the system is not significantly affected, as shown in Figure 3.1 N.