Chemotactic drift speed for bacterial motility pattern with two alternating turning events Evgeniya V. Pankratova1* , Alena I. Kalyakulina1 , Mikhail I. Krivonosov1 , Sergei V. Denisov1, 2 , Katja M. Taute3 , Vasily Yu. Zaburdaev4, 5 . 1 Institute of Information Technologies, Mathematics and Mechanics, Lobachevsky State University, Nizhniy Novgorod, Russia 2 Department of Theoretical Physics, University of Augsburg, Germany 3 Rowland Institute at Harvard, Harvard University, Cambridge, USA 4 Max Planck Institute for the Physics of Complex Systems, Dresden, Germany 5 Institute of Supercomputing Technologies, Lobachevsky State University, Nizhniy Novgorod, Russia *E-mail:
[email protected]
Supporting information S1 Appendix. Drift speed calculation The motility patterns of bacteria can be considered as a process composed of two main alternating phases of motion: “runs” and “turnings”. Although the angular changes during the turning events are stochastic, analysis of the experimental data shows that cells are able to exhibit directional persistence that can be different for various species. E. coli cells have a motility pattern with a preferential turning angle ∆ϕ that can be characterized by the parameter α = hcos ∆ϕi. V. alginolyticus bacteria perform motility with two alternating re-orientation events that can be specified by two parameters α = hcos ∆ϕ1 i and β = hcos ∆ϕ2 i. The calculation of chemotactic drift speed vd for motility with a single directional parameter was performed by Locsei [1]. Here, we extend his approach to more complicated case of bacterial chemotaxis with two alternating turning events. Note that, the algorithm presented below can be easily extended for random walks with arbitrary number of alternating turning events. Moreover, taking into account the change of parameters (mean run durations and rotational diffusion coefficients) between appointed turnings, more accurate prediction for the chemotactic drift speed can be obtained. Here, for simplicity, we restrict our analysis by assumption that all parameters for “run”-phases of the considered pattern are the same. Without loss of generality we consider a random walk whose first turning event for t = tβ > 0 is specified by the parameter β, i.e. with a turning of type α at t = 0. Let zβ be the z location of a cell at the end of a run, relative to its position at the beginning of the considered random walk. At t = tβ , the direction of motion is changed with parameter β and the cell performs a run of duration tα . To obtain the drift velocity vd in a first-order expansion with respect to |∇c|, we determine the average displacement of a run hz β i and a subsequent run hz α i. In our calculations, as in [1], we treat the duration of turning events as negligible. As the expressions for hz β i and hz α i are first order in |∇c|, the mean duration of “two runs” process is given by 2τrun , and the chemotaxis drift speed becomes vd =
hz β i + hz α i . 2τrun
(S1)
For hz α i and hz β i we calculate expectation over all possible paths taking into account that the position of the bacterium at a time t on a particular path is random.
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In the first step, we calculate the mean displacement of the run hz β i: Z ∞ hz β i = h z(t)p(t)dti.
(S2)
0
where z(t) is position of a cell at time t in particular path, and p(t) is the probability density, that a run starts at t = 0 and stops and t = tβ . Since the paths are independent of tβ , we may take the path expectation inside the integral over run times Z ∞ hz β i = hz(t)p(t)idt, (S3) 0
where p(t) is given by [1] p(t) = λ(t)e−
Rt 0
λ(t0 )dt0
.
(S4)
In accordance with the idea of de Gennes we assume that the turning rate λ (t) in (S4) in the presence of chemoattractant becomes λ(t) = λ0 [1 − ∆(t)],
(S5)
where λ0 = 1/τrun is the mean turning rate of bacteria, and the fractional change in λ (t) caused by the presence of chemicals in the environment, is: Z t ∆(t) = c(t0 )R(t − t0 )dt0 . (S6) −∞ 0
In (S6) c(t ) is chemoattractant concentration experienced by the cell at time t0 and R(t) is the cell’s response function. In the case of small chemical gradient |∇c| in z direction, the chemoattractant concentration c(t) at the cell’s position z(t) can be written in the form c(t) = c0 + |∇c|z(t).
(S7)
Substituting (S7) into (S6) yields Z
t
∆(t) =
[c0 + |∇c|z(t0 )]R(t − t0 )dt0 = c0
Z
−∞
t
R(t − t0 )dt0 + |∇c|
−∞
Z
t
z(t0 )R(t − t0 )dt0 .
−∞
(S8) Since the response function R has a zero integral, an additive constant in (S8) has no effect on ∆(t). Therefore, the equality (S8) can be rewritten in the form Z t ∆(t) = |∇c| z(t0 )R(t − t0 )dt0 . (S9) −∞
To simplify forthcoming calculations, we first consider the special case of a delta-response in time R(t) = Aδ(t − T )
(S10)
with delay time T and strength A, as it was done in [2]. In this case, the fractional change in turning rate becomes Z t ∆(t) = A|∇c| z(t0 )δ(t − T − t0 )dt0 = A|∇c|z(t − T ). (S11) −∞
Thus, taking into account that for the run lasting from t = 0 until t = tβ , the average displacement (S3) after integration by parts can be written as
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∞
Z
hvz (t)e−
hz β i =
Rt 0
λ(t0 )dt0
idt,
(S12)
0
where vz (t) = dz (t) /dt and λ (t0 ) = λ0 [1 − A|∇c|z (t0 − T )], one can obtain Z ∞ Rt 0 0 hvz (t)e−λ0 t eλ0 A|∇c| 0 z(t −T )dt idt. hz β i =
(S13)
0
For a small chemical gradient (∆(t) 1) expanding the exponential and keeping R Rt λ0 A|∇c| 0t z(t0 −T )dt0 only the first-order terms in |∇c|, i.e. e ≈ 1 + λ0 A|∇c| 0 z(t0 − T )dt0 , one obtains Z hz β i =
∞
hvz (t)ie−λ0 t dt + λ0 A|∇c|
∞
Z
e−λ0 t
Z
hz(t0 − T )vz (t)idt0 dt.
(S14)
0
0
0
t
Substitution of hvz (t)i = e−2Dr t hv(0+ )i [1] into the first integral of (S14) and using Rt z(t) = 0 vz (s)ds for the second integral of (S14) yields hv(0+ )i hz β i = + λ0 A|∇c| λ0 + 2Dr
∞
Z
e
−λ0 t
"Z
0
t
Z
t0 −T
!
# 0
hvz (s)vz (t)ids dt 0
dt.
(S15)
0
Also within the first order approximation, the velocity is governed by an isotropic distribution, which implies v02 he(s) · e(tβ )i (S16) 3 where v0 is a constant speed during a run, and the directional correlation function reads [3] hvz (s)vz (tβ )i =
−2D (t −s) r β e −2Dr tβ −(λ0 +2Dr )|s| αe e · he(s) · e(tβ )i = q √ √ √ √ β sinh α βλ |s| + cosh α βλ |s| 0 0 α
if 0 ≤ s < tβ , if s < 0. (S17)
To obtain the second line of (S17), we made a decomposition he(s) · e(tβ )i = he(s) · e(0− )ihe(0− ) · e(0+ )ihe(0+ ) · e(tβ )i,
(S18)
where the direction correlation function for directions immediately before and after the turning at t = 0 equals to α due to our choice of the considered motility pattern: he(0− ) · e(0+ )i = α,
(S19)
whereas the last multiplier in (S18) can be obtained from the Fokker-Planck equation (see S2 Appendix and [1] for details). he(0+ ) · e(tβ )i = e−2Dr tβ ,
(S20)
where Dr is the rotational diffusion coefficient describing the influence of rotational Brownian motion during the “run”-phase of bacterial swimming. To determine the first multiplier in the product (S18) we should take into account that for t < 0 the cell’s motility obeys to the pattern with two alternating turning events with additional
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reorientations due to rotational Brownian motion during the runs. For this case, the direction correlation function for the process, whose first run interrupts by a turning of type β becomes "r he(s) · e(0− )i = e−(λ0 +2Dr )|s|
# √ p √ p β sinh α βλ0 |s| + cosh α βλ0 |s| , (S21) α
see [3] for details. After inserting (S16) and (S17) into the second integral of (S15) for the mean displacement of the first run we obtain
hz β i =
√ p i h √ p hv(0+ )i v2 α βλ0 T + mβ sinh α βλ0 T + nβ , + 0 λ0 A|∇c| kβ cosh λ0 + 2Dr 3 (S22)
where λ0 e−(2Dr +λ0 )T (2Dr + λ0 + αλ0 ) 2 , (2Dr + λ0 )(4Dr2 + 4D r λ0 + (1 − αβ)λ0 ) √ λ0 e−(2Dr +λ0 )T α(2Dr + λ0 + βλ0 ) , mβ = √ β(2Dr + λ0 )(4Dr2 + 4Dr λ0 + (1 − αβ)λ20 ) 2 −λ0 α(2Dr + λ0 + βλ0 ) nβ = . (2Dr + λ0 )2 (4Dr2 + 4Dr λ0 + (1 − αβ)λ20 ) kβ =
(S23)
Proceeding in the same way as for zβ , one can obtain the mean displacement of the second run, which is interrupted by turning of type α:
hz α i =
hv(t+ β )i λ0 + 2Dr
+
√ p h √ p i v02 α βλ0 T + mα sinh α βλ0 T + nα , λ0 A|∇c| kα cosh 3 (S24)
where λ0 e−(2Dr +λ0 )T (2Dr + λ0 + βλ0 ) 2 , (2Dr + λ0 )(4Dr2 + 4D r λ0 + (1 − αβ)λ0 ) √ λ0 e−(2Dr +λ0 )T β(2Dr + λ0 + αλ0 ) mα = √ , α(2Dr + λ0 )(4Dr2 + 4Dr λ0 + (1 − αβ)λ20 ) −λ20 β(2Dr + λ0 + αλ0 ) nα = (2Dr + λ0 )2 (4Dr2 + 4Dr λ0 + (1 − αβ)λ20 ) kα =
(S25)
and v(t+ β ) is the z component of the cell’s velocity at the beginning of the second run. As before, to obtain the directional correlation function for t < tα , we should expand he(s) · e(tα )i into a product of direction correlations between different times. However, since we consider the motility with regularly alternating reorientations of bacteria, we can claim its similarity to the previous one (S17) with obvious replacement for the following parameters tβ → tα and α ↔ β. Having found the integrals in expressions for hz β i and hz α i, we turn our attention to hv(0+ )i and hv(t+ β )i. In our case, for random walk with a turning of type α at t = 0 and type β at t = tβ , using the definitions for the parameters α and β we can write he(0− ) · e(0+ )i = α, + he(t− β ) · e(tβ )i = β.
(S26)
From these definitions follows that the expected velocities immediately after the turning event can be expressed in terms of the expected velocities immediately before it (see [1] for details):
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hv(0+ )i = αhv(0− )i, − hv(t+ β )i = βhv(tβ )i.
(S27)
On the other hand, to get the expected velocity at the end of run commencing at t = 0 and interrupting at t = tβ we may take the integral Z ∞ − hv(tβ )i = hv(t)p(t)idt. (S28) 0
Substituting (S5) into (S4) and using (S11) yields p(t) = λ0 [1 − A|∇c|z(t − T )] e−λ0
Rt 0
[1−A|∇c|z(t0 −T )]dt0 .
(S29)
As before, expanding the exponential and keeping only the linear term, we can obtain Z t −λ0 t 0 0 p(t) = λ0 e 1 − A|∇c|z(t − T ) + λ0 A|∇c| z(t − T )dt . (S30) 0
hv(t+ β )i
According to (S27), we obtain the expression for hv(0+ )iλ0 (1) (1) + hv(tβ )i = β − λ0 A|∇c|(I1 − I2 ) . λ0 + 2Dr
(S31)
where (1)
I1
Z
∞
e−λ0 t hv(t)z(t − T )idt,
=
(S32)
0
and (1) I2
Z
∞
Z
∞ −λ0 t
= λ0
e
hv(t)z(t − T )idt dt0 . 0
(S33)
t0
0
Similar calculations for the second run give us the expression for the expected velocity immediately before the turning of type α at t = tα + tβ hv(t− α )i =
hv(t+ β )iλ0
(2)
(2)
− λ0 A|∇c|(I1 − I2 ). (S34) λ0 + 2Dr But, since the expected velocity before this α-turning does not change from that was for − + − α-turning at t = 0, i.e. hv(t− α )i = hv(0 )i, and since hv(0 )i = αhv(0 )i, the equation (S34) can be rewritten as " # hv(t+ β )iλ0 (2) (2) + hv(0 )i = α − λ0 A|∇c|(I1 − I2 ) . (S35) λ0 + 2Dr The equalities (S31) and (S35) give us a system of linear equations with variables hv(0+ )i and hv(t+ β )i that provide expressions for the cell’s velocities at the beginning of run after type α and type β turning events, respectively 2
βλ0 (2) (1) (1) (2) − I + I , I − I 2 2 1 1 2 (λ0 + 2Dr ) − αβλ20 λ0 + 2Dr 2 βA|∇c|λ0 (λ0 + 2Dr ) αλ0 (2) (2) (1) (1) hv(t+ )i = I − I − I1 + I2 . 2 1 β 2 (λ0 + 2Dr ) − αβλ20 λ0 + 2Dr
hv(0+ )i =
αA|∇c|λ0 (λ0 + 2Dr )
(S36)
Finally, we combine all preceding results according to (S1), and thus arrive at the chemotactic drift speed for the delta-response
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vδ =
h √ p √ p i v02 λ0 A|∇c| kδ cosh α βλ0 T + mδ sinh α βλ0 T + nδ 3
(S37)
where denoting sαβ = α + β and dαβ = −1 + αβ the coefficients kδ , mδ , nδ are:
kδ =
λ0 e−(2Dr +λ0 )T 4Dr2 (2 − sαβ ) − 8Dr dαβ λ0 − (2 + sαβ ) dαβ λ20
, 2 2 (4Dr2 + 4Dr λ0 − dαβ λ20 ) λ0 e−(2Dr +λ0 )T 4Dr2 (sαβ − 2αβ) − 4Dr dαβ λ0 sαβ − λ20 dαβ (sαβ + 2αβ) mδ = , 2√ √ 2 (4Dr2 + 4Dr λ0 − dαβ λ20 ) α β λ2 4Dr2 (−sαβ + 2αβ) + 4Dr λ0 dαβ sαβ + λ20 dαβ (sαβ + 2αβ) , nδ = 0 2 2 (2Dr + λ0 ) (4Dr2 + 4Dr λ0 − dαβ λ20 ) (S38) Having found vδ , we obtain the chemotaxis drift speed for the response function " 2 # λ0 T λ0 T −λ0 T R(T ) = W λ0 e 1− − , (S39) 2 2 where W is a single normalization constant with the dimension of volume, according to Z ∞ vδ (t) dT, (S40) vd = R(T ) A 0 that yields P7 v02 λ20 W |∇c| j=0 aj (α, β)Dr7−j λj0 , vd = P10 4 j=0 bj (α, β)Dr10−j λj0
(S41)
where a0 (α, β) = −256 (−2 + sαβ ), a1 (α, β) = −64 (−46 + 12αβ + 17sαβ ), a2 (α, β) = −32 (−222 + 116αβ + (51 + 2αβ)sαβ ), a3 (α, β) = 16 582 − 448αβ + 20α2 β 2 − (48 + 29αβ)sαβ , a4 (α, β) = 8 894 − 896αβ + 108α2 β 2 + (63 − 126αβ + 10α2 β 2 )sαβ , a5 (α, β) = −4 −804 + 994αβ − 222α2 β 2 + 4α3 β 3 + (−183 + 250αβ − 53α2 β 2 )sαβ , a6 (α, β) = −2 −392 + 590αβ − 194α2 β 2 − 4α3 β 3 + (−152 + 243αβ − 97α2 β 2 + 6α3 β 3 )sαβ , a7 (α, β) = −d2αβ −80 − 12αβ + 4α2 β 2 + (−44 + 7αβ)sαβ ,
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and b0 (α, β) = 1024, b1 (α, β) = 8192, b2 (α, β) = 29184 − 1280αβ, b3 (α, β) = 60928 − 8192αβ, b4 (α, β) = 82496 − 22784αβ + 640α2 β 2 , b5 (α, β) = 75648 − 35968αβ + 3072α2 β 2 , b6 (α, β) = 47552 − 35248αβ + 6144α2 β 2 − 160α3 β 3 , b7 (α, β) = 20224 − 21952αβ + 6560α2 β 2 − 512α3 β 3 , b8 (α, β) = 5568 − 8480αβ + 3948α2 β 2 − 624α3 β 3 + 20α4 β 4 , b9 (α, β) = 896 − 1856αβ + 1272α2 β 2 − 344α3 β 3 + 32α4 β 4 , b10 (α, β) = 64 − 176αβ + 172α2 β 2 − 73α3 β 3 + 14α4 β 4 − α5 β 5 . If we assume, that all turning angles are equal (α = β), we obtain the following expression vd = W |∇c|v02
λ20 (1 − β)(4Dr + λ0 (5 − 2β)) , 6(2Dr + λ0 − βλ0 )(2Dr + (2 − β)λ0 )3
(S42)
which agrees with formula (27) in [3]. If we assume, that the cell’s motion pattern is “run-tumble-flick” (α = 0), we get W |∇c|v02 λ20 · 192(Dr + λ0 )4 (2Dr + λ0 )2 2 3 (16Dr (2 − β) + 4Dr λ0 (22 − 5β) + 2Dr λ20 (38 + 5β) + λ30 (20 + 11β)), vd =
(S43)
which also agrees with formula (28) in [3].
References 1. Locsei JT. Persistence of direction increases the drift velocity of run and tumble chemotaxis. J Math Biol. 2007; 55:41–60. 2. de Gennes PG. Chemotaxis: the role of internal delays. Eur Biophys J. 2004; 33: 691–693. 3. Taktikos J, Stark H, Zaburdaev V. How the Motility Pattern of Bacteria Affects Their Dispersal and Chemotaxis. PLoS ONE. 2013; 8(12): e81936.
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