Intuitive Energy Landscape model - Appendix to Chapter 2

5.2 Appendix to Chapter 2

5.2.2 Intuitive Energy Landscape model

Assumingbis long enough thatb−1≈b,

k_eff(0)≈(2e^−|∆G^bp^|/RT)k_eff(1) . (5.1) Choosingk_uni. We choosek_uni such that the rate of dissociation of the last base pair of the toehold is approximately equal to the rate of fraying of every other base pair. So, we set

k_unie^−|∆G^bp^|/RT ≈k_bie^(−|∆G^bp^|+|∆G^assoc^|)/RT u₀, (5.2) which yields

k_uni≈k_bie^|∆G^assoc^|/RT u₀ (5.3)

whereu0= 1M is the reference concentration.

Modelingk_eff. Conceptually, we may split the kinetics of strand displacement into two parts:

the concentration-dependent, bimolecular part involving collision and formation of the first base pair of the toehold (state A to B) and the unimolecular part comprising the formation of the remaining base pairs of the toehold (states B to C) and branch migration (states C to F). The unimolecular part is modeled as a random walk on the free energy landscape starting at B, and ending either at A (no displacement) or at E (successful displacement). In the low-concentration regime, the unimolecular part finishes (one way or the other) much faster than the rate of collision; so, we model the effective rate constant of strand displacement,k_eff, as

k_eff=kv×p (5.4)

wherekvis the rate constant for the formation of the first toehold base pair andp=P(reaching E before A starting at B) is the probability of successful displacement starting at B. Since the IEL assumes a collision rate of

k_bi×u,kv=k_bi.

Calculatingp. First, given the rates of the individual transitions as specified in the kinetic model, the probabilities of transition from each state to its neighbors can be calculated. Letpi,jbe the probability of going from stateito statej. Then, we choose

pi,i+1= k_i,i+1

ki,i+1+k_i,i−1 , (5.5)

pi,i−1= k_i,i−1

ki,i+1+k_i,i−1 . (5.6)

Since we are interested in the probability that the random walk finishes at one end rather than the

A 1 2 4 6 8 10 12 14 16 18 20 ï

ï 0 10

Number of strand displacement steps completed

∆ G° (kcal/mol)

Free energy landscape for toehold length 0

B’

F A

A’

∆ G

box

(k cal / mol )

Figure 5.2: IEL free energy landscape at 25^◦Cfor a 0-base toehold. First, the invader and the substrate- incumbent complex are unconstrained by each other (A). Then, the substrate-incumbent complex frays at one end (A⁰). This fraying step, at the cost of one base pair stack (|∆G_bp|), is necessary to enable a collision leading to a base pair, since no toehold is present. Once such a collision occurs, the first base of the incumbent has effectively been displaced by the invader (B⁰). The remaining states are identical to the positive toehold case.

other, we use absorbing boundaries:

pi,i= 1fori= A, E (A⁰for 0 toehold). (5.7) Given these transition probabilities,pcan be calculated analytically using the method of Rudolph [173] for calculating absorption probabilities for a 1-dimensional random walk with absorbing boundaries.

0 5 10 15

0 2 4 6 8

Toehold length log 10(k eff)

IEL with a 2 RT (0.7 stacks) plateau

Metropolis Kawasaki

0 5 10 15

0 2 4 6 8

Toehold length log 10(k eff)

Original Intuitive Energy Landscape (IEL)

Metropolis Kawasaki

Aï 0 4 8 12 16

ï ï ï ï ï

Number of strand displacement steps

∆ G° 25 (kcal/mol)

Sequence dependent landscape (toehold 6)

Zhang & Winfree (2009) Zhang & Winfree (2009)

Zhang & Winfree (2009)

A B

C D

0 5 10 15

0 2 4 6 8

Toehold length log 10(k eff)

Sequence dependent landscape

Metropolis Kawasaki A

E F

Figure 5.3:Strand displacement kinetics predicted by the IEL under varying conditions, for both Metropolis and Kawasaki unimolecular rate models. (A) IEL (2.6, 0) (B) IEL (2.6, 1.2) (D) for the sequence dependent free energy landscape predicted by the NUPACK energy model with Dangles = “Some” (shown in (C)). None of these variations is consistent with experimental data [147]. (C) States A–F pertain to Figure2.3in the main text. The sequence dependent landscape is much “rougher” than the IEL, but this “roughness” goes no further in accounting for the data (D).

Alternate unimolecular rate model. We also considered the Kawasaki [174] method for as- signing unimolecular transition rates in the IEL:

kij=k_uni×e^−(∆G^box^(j)−∆G^box^(i))/2RT (5.8) k_ji=k_uni×e^−(∆G^box^(i)−∆G^box^(j))/2RT (5.9) As with the Metropolis rate model, this choice guarantees detailed balance and thus that the sys- tem eventually converges to a Boltzmann distribution:

Prob(i) = 1 Qe⁻

∆Gbox(i)

RT where Q=X

e⁻

∆Gbox(i) RT .

Once again, we choosek_unisuch that the rate of dissociation of the last base pair of the toehold is approximately equal to the rate of fraying of every other base pair. For the Kawasaki method, we have

k_unie^−|∆G^bp^|/2RT ≈k_bie^(−|∆G^bp^|+|∆G^assoc^|)/RT u₀, (5.10) which yields

k_uni≈k_bi×e^(−|∆G^bp^{|/2 +}^|∆G^assoc^|)/RT u0. (5.11) Withk_uni≈1.8×10⁷/s as calculated above, the Kawasaki method consistently predicts a slightly smaller acceleration ink_effwith toehold length than the Metropolis [172] method (see Figure5.3);

therefore our conclusions about constraints on IEL parameters that are necessary to match experimental data are robust to this choice. In the analysis below, the Metropolis method is assumed throughout.

Analytical approximations fork_eff(h).The IEL is simple enough to yield itself to an approximate analysis that permits intuitive understanding of the model. Forh >1, we may approximate k_eff(h)as:

k_eff(h)≈k_bi×p_zip×p_bm|toe(h) (5.12) where p_zip is the probability that the full toehold “zips up” once the first base pair is formed, k_bi×p_zipis the bimolecular rate constant for all the base pairs in the toehold to form andp_bm|toe(h) is the probability that, once the toehold has zipped up, the incumbent is displaced before the toehold dissociates.

Forh= 1, there are no other bases in the toehold to zip up, sok_eff(1)may simply be approxi-

mated as

k_eff(1)≈k_bi×p_bm|toe(1). (5.13) We shall now estimatep_bm|toe(h). Letk_firstbe the rate at which the first base of the incumbent is displaced by the invader, once the toehold is bound. Letk_r(h)be the rate at which the toehold of lengthhdissociates from the state in which it is fully bound (i.e. state C of Figure2.3in the main text). We estimatekfirst andkr(h)later. The probability of displacing the first base of the incumbent before the toehold dissociates is simplykfirst/(kfirst+kr(h)).

For simplicity, we assume that once(b−1)bases of the incumbent have been displaced, the last base always dissociates. This approximation allows us to think about branch migration as a flat random walk. Our approximation is reasonable because the probability that the invader dissociates without completing strand displacement after displacing(b−1)bases of the incumbent can only be 0.5, in the worst case (for a 0-base toehold, withk_uni >> k_bi), and hence introduces negligible error on a logarithmic scale. Moreover, if branch migration is indeed significantly slow relative to fraying, we would expect that the last few bases of the incumbent would fray, leading to the dissociation of the incumbent even before branch migration completes. Once the first base of the incumbent has been displaced, the probability of displacing the remaining bases of the incumbent before going back to the toehold-only-bound state is1/(b−1), according to the gambler’s ruin analysis (see Section 14.2 of Feller [169]). With a probability of(b−2)/(b−1), we return to the toehold bound state - from where, by definition, displacement succeeds with a probability of p_bm|toe(h).

Therefore, we havep_bm|toe(h)≈ kfirst

k_first+k_r(h) 1

b−1+b−2

b−1×p_bm|toe(h)

(5.14) which yields

p_bm|toe(h)≈ k_first

kfirst+ (b−1)kr(h). (5.15) Therefore, equation5.12yields

k_eff(h)≈k_bi×pzip× k_first

kfirst+ (b−1)kr(h) . (5.16) With some algebra, equation5.16yields

k_eff(h)≈ k_bi×pzip

1 + (b−1)^k_k^r^(h)

first

(5.17)

forh >1.

Analyzing equation 5.17 provides intuition about the dependence ofk_eff(h) onh. For long toeholds, i.e. in the saturation limit,(b−1) ^k_k^r^(h)

first <<1; this yields the saturation value

k_eff(∞)≈k_bi×pzip. (5.18)

For short toeholds, i.e in the “sloping” part of thek_eff(h)curve,(b−1)^k_k^r^(h)

first >>1and this yields k_eff(h)≈ k_bipzipkfirst

(b−1)kr(h). (5.19)

We now estimatep_zip. Forh >1, we assume, for simplicity, that the formation of a second base pair of the toehold will guarantee zippering. Then,pzipmay be approximated as

pzip≈ k_uni

k_uni+k_r(1). (5.20)

Note thatk_r(1)is just the rate for the simple bimolecular dissociation step, and so we have

k_r(1)≈k_biλ (5.21)

whereλis defined as

λ=e^−(|∆G^bp^|−∆G^assoc^)/RT u₀. (5.22) We now estimatekr(h)forh > 1. Once a toehold of lengthhis fully bound,(h−1) uphill fraying steps and 1 dissociation step need to occur for the toehold to dissociate. Therefore,

kr(h)≈k_fray(h−1)×(1−pzip) (5.23) wherek_fray(h−1)is the rate at which(h−1)base pairs of the toehold fray, starting from the toehold fully bound state, to leave the invader bound by 1 base of the toehold (state C to B in Figure2.3in the main text).

Clearly,

k_fray(h−1) =k_unie^−(h−1)^|∆G^bp^|/RT. (5.24) From equations5.20,5.23and5.24, we get

kr(h)≈e^−(h−1)^|∆G^bp^|/RT × 1

1/k_uni+ 1/kr(1) . (5.25) We also recall that in our particular parameterization we chosek_uni such that kr(1) ≈ k_fray(1), although we do not use or need that condition in this analysis.

Lastly,

kfirst=1

2 ×k_uni×e^−∆G^s+p^/RT (5.26)

where∆G_s+pis defined as

∆G_s+p= ∆G_s+ ∆G_p (5.27)

and∆Gsis the sawtooth amplitude and∆Gpis the plateau height.

Substituting forkr(h)from equation5.25,kfirstfrom equation5.26andpzipfrom equation5.20 into equation5.19, we may approximatek_eff(h)forh > 1in the pre-saturation regime. We also assume thatbis long enough thatb−1≈b. With some algebra, we get

k_eff(h)≈k_unie^−(∆G^s+p^−h|∆G^bp^|)/RT η

2b (5.28)

whereηis defined as

η= 1

e^∆G^assoc^/RT u₀. (5.29)

We now approximatek_eff(1)using equation5.13. Using equations5.15,5.21and5.26, we get k_eff(1)≈ ρ k_bi

ρ+ 2b λ e^∆G^s+p^/RT (5.30)

whereρis defined as

ρ=k_uni/k_bi. (5.31)

Having already assumed thatb−1 ≈b, we may approximatek_eff(0)in terms ofk_eff(1)using equation5.1. Along with equation5.30, this yields

k_eff(0)≈ 2e^−|∆G^bp^|/RT ρ k_bi

ρ+ 2b λ e^∆G^s+p^/RT . (5.32)

Although equations5.30and5.32seem different in form from equation5.28, that is only because equation5.28assumes the pre-saturation (“sloping”) regime, whereas equations5.30and 5.32 are more general. For reasonable values ofρ and long enough b, such as the default IEL values (ρ≈25, b= 20), we may use the approximation

ρ+ 2b λ e^∆G^s+p^/RT ≈2b λ e^∆G^s+p^/RT. (5.33) This yields

k_eff(1)≈k_unie^−(∆G^s+p^−|∆G^bp^|)/RT η

2b (5.34)

and

k_eff(0)≈k_unie^−∆G^s+p^/RT η

b (5.35)

which are consistent in form with equation5.28, but for the factor of two forh= 0. We define

A_h₂_,h₁ = log₁₀(k_eff(h₁))−log₁₀(k_eff(h₂)) (5.36) to be the orders of magnitude acceleration ink_effas toehold length increases fromh1toh2. We now use equations5.18and5.32to study the dependence ofA∞,0onρ,∆G_s+pandb. With some algebra, equations5.18and5.32yield

A∞,0≈µ+ log₁₀

ρ+ 2b λ e^∆G^s+p^/RT ρ+λ

(5.37) whereµis defined as

µ= log₁₀

e^|∆G^bp^|/RT/2

≈0.9. (5.38)

Dependence onρ=k_uni/k_bi.To analyze the effect of changingρonA_∞,0, let us choose a priori reasonable values for the other parameters:b= 20,∆Gs= 1.5 stacks (2.6 kcal/mol) and∆Gp= 0.

Then, equation5.37becomes

A∞,0≈0.9 + log₁₀

ρ+ 4000 ρ+ 1.4

(5.39)

whose behavior is qualitatively similar to the numerical curve plotted in Figure5.4. Moreover, the difference betweenA_∞,0atρ=∞andρ= 0as predicted by equation5.39is approximately 3.5, which is consistent with the total variation inA15,0as numerically plotted in Figure5.4.

Dependence on∆Gs+p. For the defaultk_uni andk_bivalues used in the IEL,ρ/(ρ+λ)may be neglected in equation5.37to yield

A∞,0≈µ+ log₁₀

2b e^∆G^s+p^/RT λ ρ+λ

. (5.40)

This yields an approximately linear dependence on∆Gs+p:

A_∞,0≈c1+c2∆Gs+p (5.41)

where

c1=µ+ log₁₀ 2b λ

ρ+λ

(5.42)

ï 1 0 10 2

3 4

log

(k

uni

/k

)

Orders of mag. acceleration

IEL(2.6, 0)

Figure 5.4:The influence ofkuni/k_bion the orders of magnitude acceleration ink_effbetween toehold lengths 15 and 0 (A15,0) predicted by IEL(2.6, 0). The red circle indicates the default value ofk_uni/k_bi. The IEL exhibits sigmoidal response tolog₁₀(kuni/k_bi)and even implausibly low values ofkuni/k_bido not result in acceleration that matches the experimentally observed value of 6.5 orders of magnitude.

and

c2= 1

RTlog_e(10). (5.43)

100 500 1000 2000

6 6.5 7 7.5 8 8.5

Branch migration length

Orders of magnitude acceleration

Figure 5.5:Orders of magnitude acceleration ink_effbetween toehold lengths 15 and 0 (A15,0), as a function of the length of the branch migration domain, as predicted by IEL(5.3, 2.0).

Dependence onb.From equation5.40, we have

A_∞,0≈log₁₀(b) +c3 (5.44)

where

c₃=µ+ log₁₀

2e^∆G^s+p^/RT λ ρ+λ

. (5.45)

The logarithmic dependence ofA∞,0onbis qualitatively similar to the numerical curve ofA15,0

plotted in Figure5.5.

Estimating branch migration step times. In order to compare with experimentally inferred values [165,166], we define average branch migration step time (t_bm) as the average time taken for a single step of branch migration in either direction. For the IEL and AEL,t_bmis calculated as

tbm= 1

k_bm (5.46)

where

k_bm =k_uni×e^−∆G^s^/RT. (5.47)

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