Signal focusing through active transport

3.1 MB·English

by  Aljaz Godec

Signal focusing through active transport
Aljaˇz Godec1,2 and Ralf Metzler1,3
1Institute of Physics & Astronomy, University of Potsdam, D-14476 Potsdam-Golm, Germany
2National Institute of Chemistry, 1000 Ljubljana, Slovenia
3Department of Physics, Tampere University of Technology, FI-33101 Tampere, Finland
In biological cells and novel diagnostic devices biochemical receptors need to be sensitive to
extremely small concentration changes of signaling molecules. The accuracy of such molecular
signaling isultimately limited bythecountingnoiseimposed bythethermaldiffusionof molecules.
Many macromolecules and organelles transiently bind to molecular motors and are then actively
transported. Wehereshowthatarandomalbeitdirecteddeliveryofsignalingmoleculestowithina
5 typicaldiffusiondistancetothereceptorreducesthecorrelationtimeofthecountingnoise,effecting
1 an improved sensing precision. The conditions for this active focusing are indeed compatible with
0 observationsinlivingcells. Ourresultsarerelevantforabetterunderstandingofmolecularcellular
2 signaling and the design of novel diagnostic devices.

n
PACSnumbers: 87.15.Ya,87.15.Vv,87.16.Xa
a
J

3
Cellular signaling relaying external or internal bio- cal signaling [4, 5, 7–10] along with additional features
1
chemicalcuestypicallyinvolveslowcopynumbersofmes- such as receptor cooperativity [3] and facilitated diffu-

] senger molecules, inevitably effecting appreciable fluctu- sion [12]. Various experiments suggest that cells indeed
C
ationsinthecountofmolecularbindingeventsatspecific operate very close to the fundamental accuracylimit [1].
S receptors [1–12]. A similar limitation by counting noise
Here we extend the result (1) to the case when the
o. is encountered in modern microscopic diagnostic devices signaling molecules are not only freely diffusing in the
bi to which sensitivity is a key factor [13]. Modern micro- cell but actively transported along cellular filaments by
scopic techniques reveal molecular signaling events and
- motorproteins[20,21]effectingintermittentballisticex-
q underline their inherent stochasticity in living cells [14–
cursions [22–25]. Such an additional active component
[
17]. Nevertheless molecular signaling pathways in bio-
occurs when extracellular signaling molecules are taken

1 logical cells operate at remarkable precision [18, 19]. up by the cell via endocytosis: the molecules are en-
v The first heuristic argument about noise limitation to
gulfed into submicron lipid vesicle and then intermit-
1 biological concentration measurements is due to Berg
tently transported through the cell by motors [26]. A
4
andPurcellassumingbiochemicalreceptorsto countthe
9 similarcombinationoffreediffusionandactivetransport
number N of specific molecules in a volume equal to
2 occurswhenvirusparticlesinvadealivingcell[27]. How-
0 their linear dimension a [2]. N is then limited by Pois- ever,evenfreemoleculessuchasmessengerRNAmayat-
. sonian noise δN N 1/2. The time between two in-
1 ∼ h i tach to motors [28], or proteins move in directed fashion
0 dependent measurements is set by the time τD a2/D due to cytoplasmic drag [29]. To incorporate the active
∼
5 needed to clear the volume by diffusion, D being the componentweemploythetheoryofrandomintermittent
1 molecular diffusivity. Averaging over a time τ thus
m searchfor hidden targets [30] whichwas recently usedto
v: allows to Nm τm/τD independent measurements, re- analyze reaction kinetics in active media [31]. We show
∼
Xi ducing the noise by the factor Nm1/2. The relative accu- that active transportenables both faster as well as more
racy to measure a background concentration c is thus accurate sensing: an active noise floor exists, but it can
r h i
a δc2/ c 2 (Da c τm)−1 [2]. When the additional bind- be significantly lower than the purely diffusive counting
h i ∼ h i
ing dynamics to the biochemical receptor is explicitly noise (1). This active focusing reduces the noise corre-
taken into account, this relative error becomes [1] lation time and enables the receptor to detect relative
changes in concentration with higher accuracy. Our re-
δc2 2 1
= + . (1) sults also have directimplications to the designof active
c 2 k c (1 n )τ πDa c τ
on m m components in microscopic synthetic diagnostic systems
h i h i −h i h i
based on molecular signals [13].
The first contribution stems from the Markovian
(un)binding to the receptor at detailed balance condi- Model. We considera signalingparticle (vesicle,virus,
tions with binding rate k and average receptor occu- mRNA, or protein) in 3-dimensional cellular space, ran-
on
pancy n . The second term is the diffusional noise, up domlyswitchingbetweenapassivediffusionphasepwith
h i
to the factor π identical to the result by Berg and Pur- diffusivity D and an active ballistic phase a with veloc-
cell [2]. The prefactor of the diffusive term in Eq. (1) ity v(Ω) of constant magnitude v = v [32] in the di-
| |
was recently refined heuristically [11]. Inspired by the rection of the solid angle Ω following a Markovian dy-
early ideas of Berg and Purcell a number of studies un- namics (Fig. 1). Assuming ideally disordered cytoskele-
raveled the crucial role of diffusional noise in biochemi- tal filament orientations, the spatial direction of active
2
Speed. We assume that the system equilibrates on a
timescaleoverwhichthesignalingmoleculesmoveadis-
tance L of the order of the cell size (or that of a cellu-
lar compartment). At this stage we neglect the analyte-
receptorbindingdynamicsandadoptaprobabilisticview
of Eqs. (3). The equilibration time τ is then defined by
i
the mean squared displacement (MSD), r(τ )2 = L2.
i
h| | i
The exact result is (see Supplementary Material, SM)
h|r2(t)|i=2((vτa)2e−τta − (vτ2a−+1+3Dτp−τp−1)12e−t(ττaa+τpτp)
τp−1(vτa)2+3D 3Dτp (vτa)2(1+2τp/τa)
+ t+ − .(4)
1+τ /τ (1+τ /τ )2
a p p a )
FIG. 1: Schematic of the model system: a) Each signaling
Overaperiodofdurationτ +τ ,duringwhichthedirec-
particle performs passive thermal diffusion (blue phases) in- a p
terrupted by active ballistic excursions with constant speed tional persistence in the active phase causes a nonlinear
andrandomdirection(redphasesmovingalongtheblackmo- time dependence of r2(t) and hence a local violation
h| |i
tortracks). Theduration of both phases isdistributed expo- ofthe centrallimit theorem, aneffective diffusive regime
nentiallywithmeantimesτp,a. Whentheparticlereachesthe r2(t) t is established. Eq. (4) is a transcendental
receptor(greensphere)it bindswithon-ratekon anddissoci- he|quatio|in≃for τ , essentially depending on only three pa-
ates with rate koff. b) Magnification of thereceptor region. i
rameters: the typical distance covered in the active and
passivephases,x =vτ andx = Dτ ,andtheP´eclet
a a p p
number Pe=Lv/D. To estimate the efficiency of active
motion events is uniformly distributed. The duration of p
trafficking with respect to diffusion we compare τ with
active/passive phases is exponentially distributed with i
the purelypassiveequilibrationtime τ L2/(6D). The
meanτa,p [33]. Theconcentrationsoffreelydiffusingand 0 ≡
results for various Pe values (see Fig. 2e)) typical for
motor-bound signal particles are c (r,t) and c (r,Ω,t).
p a
biological systems are shown in Fig. 2a)-d).
A receptor with radius a is placed at r . Then the frac-
0
Active transport is more efficient for larger particles
tionaloccupancyn(t)ofthe receptorbya signalparticle
(small D) in larger domains, a direct consequence of
evolvesaccordingtoa meanfieldkinetic scheme obeying
the finite motor velocity and instantaneous direction-
detailed balance with on/off rates k ,
on/off
ality of active motion. Namely, in terms of the MSD
dn(t)/dt=k c (r ,t)(1 n(t)) k n(t). (2) diffusion and active motion display different time scal-
on p 0 off
− − ing ( t versus t2): considering only pure passive
≃ ≃
Assuming that the particle (un)binds to (from) the re- and active motion for Pe < 6 diffusion is more effi-
ceptor only from (to) the passive mode [31], the coupled cient. In the intermittent case the motion has a tran-
set of equations for the concentrations c reads sient period of duration τ +τ , which corresponds to a
a,p a a
parameter-dependent combination of both regimes dur-
∂cp(r,t) =D 2c + cadΩ cp δ(r r )dn(t), ing the relaxation towards the equilibrium partitioning
p 0
∂t ∇ τ − τ − − dt between phases a and p. After this transient period an
Z a p
∂c (r,Ω,t) c c effective diffusive regime is established with diffusivity
a = r (v(Ω)ca) a + p . (3) D =(D+x2/(3τ ))/(1+τ /τ ),whichmayormaynot
∂t −∇ · − τa 4πτp eff a p a p
be larger than D. τ can thus be smaller or larger than
i
Thesignalingtypicallyoccursintwostages. Intheini- τ . Traffickingofvesicleswith D .10 2µm2/stherefore
0 −
tial phase a change in the concentration of the signaling mostlyprofitsfromactivemotion,whereasactivemotion
particlesoccurseitherbyexchangewiththeextracellular of smaller proteins with D 10µm2/s will only be more
∼
space [19, 34] or by variation of the production and/or efficient over large distances as in eukaryotic cells (espe-
degradationrates. Uponre-equilibration(assumedto be ciallyforneurons),andonlyifaccompaniedbysignificant
muchfasterthanthemeasurementtimeτ )thereceptor phasesofpassivediffusion. Theobservedfeaturesexplain
m
readsouttheconcentrationoverthe timeτ inthemea- why it is profitable for a cell to use active transport for
m
surement phase. Inadiagnosticdeviceequivalentphases trafficking of larger particles [18, 35], despite demanding
will be observed after sample immersion and during de- more cellular resources. Similarly, active diagnostics [13]
tection periods. In an optimal signaling setup equilibra- can be faster and hence allow for a higher throughput.
tion shouldbe as fastas possible while the measurement Precision. Since the precision of the receptor mea-
phase should be as precise as possible. We now quantify surement of the signal molecule concentration should
the speed and precision of the two signaling phases. be maximized we consider small deviations from the
3
a) 100 Pe=104 τ0/τi105 b) 100 Pe=102 τ0/τi103 c) 100 Pe=1 τ0/τi101
10-1 104 10-1 102 10-1 100
10-2 103 10-2 101 10-2 10-1
L L L
x/p10-3 102 x/p10-3 100 x/p10-3 10-2
10-4 101 10-4 10-1 10-4 10-3
10-5 100 10-5 10-2 10-5 10-4
10-5 10-4 10-3 10-2 10-1 100 10-5 10-4 10-3 10-2 10-1 100 10-5 10-4 10-3 10-2 10-1 100
xa/L xa/L xa/L
d) 100 Pe=10-1 τ0/τi101 e)102 Pe for active transport of biological cargo 104
10-1 100 eukaryotic cells cargo size 103
10-1
x/Lp1100--32 10-2 µL [m]101 vesicles large proteins small proteins 110012
10-3
10-4 10-4 prokaryotic cells 100
10-5 10-5 100 10-1
10-5 10-4 10-3 10-2 10-1 100 10-2 10-1 100 101
xa/L D [µm2/s]
FIG.2: a)-d)Equilibration timesτ0/τi forvariousP´eclet numbersPefrom numericalinversionofEq.(4). Yellowlinesdenote
τ0/τi = 1. When τ0/τi > 1 active trafficking is more efficient. Note the different scales for τ0/τi. e) Pe as function of cell
(domain)sizeLanddiffusivityD forbiological cells[19,34]. Arrowsdenoteincreasingparticlesizefromright(smallproteins)
to left (large vesicles) and cell size from bottom (prokaryotic cells) to top (eukaryotic cells). The border between prokaryotic
and eukaryotic cells is around 10µm. For Pe ∼ 104 (vesicles or large proteins in eukaryotic cells), active transport always
improvesthespeed. Forsmallproteinsineukaryoticcellsandlargeproteinsinprokaryoticcells(10.Pe.102),inadditionto
largex alargex isnecessaryforactivetransporttobemoreefficient. ForPe.10−1 activetransportisalwayslessefficient.
a p
equilibrium values n = n + δn and c = c + In the biologically more important situation x a,
p,a p,a a
h i h i ≫
δc [1], taking into account the detailed balance for
thpe,abinding/unbinding transitions via kon cp /koff = Λ(xa,xp)∼1−tan−1(πxp/a)/(πxp/a), (7)
h i
exp(F/[kBT]), where F is the binding free energy [36]. whichhasthelowerboundΛ (πx /a)2[1 (πx /a)2]
min p p
SolvingEqs.(3)andfollowingtheideasofRef.[1]weuse ∼ −
asx /a 0. Thismightsuggestanapproachtowardsan
p
the fluctuation-dissipation theorem to obtain the power →
infinite absolute precision of the transportterm as x
p
spectrum of the concentration fluctuations. Within the →
0. However, at fixed total concentration c of signaling
tot
linear response regime and for receptor measurement
particlesin the transportcontrolledregimewe havethat
times τ exceeding any correlation times, the relative
m c = c /(1 + τ /τ ) corresponding to c 0 as
p tot a p p
error for the concentration measurement is [36] h i h i →
x 0, hence diverging relativefluctuations. Therefore,
p
→
there still exists a noise floor to active sensing but it can
δc2 2 Λ(x ,x )
p = + a p . (5) be significantly reduced as explained below.
c 2 k c (1 n )τ πDa c τ
h pi onh pi −h i m h pi m We gaugethe sensingprecisionatequalctot andequal
c , corresponding to the lower and upper bounds for
Compared to Ref. [1] for passive diffusion our result (5) h pi
the gain of active focusing. For equal c we compare
differsbythedimensionlessfactorΛ(x ,x )inthesecond tot
a p the relative accuracy of measuring different concentra-
term, for which always Λ(x ,x ) 1 holds [36].
a p ≤ tions of freely diffusing molecules in active and passive
We now focus on the transport controlled regime in
sensing,finding thatthe precisionis alwaysworseforac-
which k c ,k τ 1,τ 1 and k c /k 1.
onh pi off ≫ a− p− onh pi off ≫ tive transport as compared to free diffusion (Fig. 3a)-c)
Hence we neglect the first term in Eq. (5) and consider
and asymptotic results in e) and f)) and becomes worse
the remaining active noise floor. If the active excursions
with longeractive excursions. The reasonis that despite
are short compared to the receptor size, x /a 1,
a ≪ reducingtheabsolutefluctuationsbyactivedynamicswe
Λ(x ,x ) √5axptanh−1 (√π5xaax)p2 1+ 31(xa/xp)2 −21 . atiroensm.eCaosunrvienrgseslym,aFlliegrs.a3ndd)-fs)msahlolewrsetffheacttiivfewceocnocmenptarrae-
a p ∼ (πxa)2 (cid:16)[1+ 13((cid:2)xa/xp)2]1/2 (cid:3) (cid:17) the precisionatidenticalconcentrationctot(1+τa/τp) of
(6) freemoleculestheaccuracycanbeimprovedsignificantly
4
a) 105 σa/σ1003 b) 105 σa/σ0 c) 105 σa/σ0
104 6D/va=1 104 6D/va=102 103 104 6D/va=104 104
103 102 103 103 103
a102 a102 102 a102
x/p101 x/p101 x/p101 102
100 101 100 101 100 101
10-1 10-1 10-1
10-2 100 10-2 100 10-2 100
10-210-1100101102103104105 10-210-1100101102103104105 10-210-1100101102103104105
xa/a xa/a xa/a
d) 105 σa/σ1000 e) 105 xp=0.05 xp=5·104 f) 105 xa=0.05 xa=5·104
104 same concentration 104 66DD//vvaa==1102 104 66DD//vvaa==1102
6D/va=104 6D/va=104
103 103 103
x/ap110012 10-1 σσ/a0 110012 σσ/a0 110012
100 100 100
1100--21 1100--21 ((sscc)) xxpp==05.·01504 1100--21 ((sscc)) xxaa==05.·01504
10-210-1100101102103104105 10-210-1100101102103104105 10-210-1100101102103104105
xa/a xa/a xp/a
FIG. 3: a)-c) Fractional variance σ = [δc2/hc i2]1/2 of concentration fluctuations at the receptor site for active trafficking
a p p
compared to thermal diffusion, σ0 = [πDactotτm]−1/2 as function of the average distance traveled in both phases in units of
the receptor radius a. In a)-c) the total concentration ctot is kept constant, hence decreased sensing accuracy is solely due to
exceedinglysmallequilibriumconcentrationsinthepassivephase. d)σa/σ0 forequalequilibriumconcentrations(scinpartse),
f))inthepassivephasehcpifordifferenttotalconcentrations=ctot(1+τa/τp): astrongincreaseinsensingaccuracyisobserved
for xp/a → 0. a)-d) use the full results (5) and (S7) in SM [36]. e) Horizontal cross-sections of σa/σ0 at large and small xp
(symbols)compared totheapproximations(6),(7). f) Verticalcross-sections at largeandsmall x (symbols)compared tothe
a
approximations (6),(7). Notethedifferentscales forσa/σ0 andtheexcellent agreement betweenfull andapproximateresults.
The non-monotonicity at small x is duetothe interplay of signal focusing (smaller Λ) and decreasing hc i.
p p
as long as x . a (otherwise, in this regime active traf- pounds in the case when an active transport component
p
ficking does not affect sensing precision). The ingredi- is present. This occurs for various signaling particles
entsnecessarytounderstandthisreducedcountingnoise (proteins,mRNAmolecules,vesiclescontainingsignaling
are: (i) only fluctuations on a scale a are relevant for molecules, or viruses) by direct shuttling of these signal-
∼
the sensingaccuracyand(ii) the receptoronly’sees’free ing cues by molecular motors or by cytoplasmic drag.
particles. Hence, at finite temperatures perfect signaling
Compared to the purely diffusive signaling considered
corresponds to the situation in which upon release from
so far we showed that the counting noise (the limit to
the receptor the particles immediately bind to a motor
accurate receptor measurement) for active sensing can
and are swept away over distances > a. Concurrently
becomesignificantlyreduceddue toactivefocusing. The
diffusive displacements must be . a to assure focused
only contributions to the counting noise stem from par-
delivery in the sense that any unbinding form the motor
ticles,whichareactivelytransportedtowithintheparti-
only contributes if it occurs at r r . a. This re-
| − 0| cle’s typical free diffusion distance to the receptor. This
duction of local concentration fluctuations by means of
reduces the correlation time of the receptor occupancy
intermittentactiveexcursionsisexactlythe active signal
noiseandrenderstheaveragingoverameasurementtime
focusing mentioned above.
τ more efficient. The importance of active signaling in
m
Conclusion. The last years have seen significant ac- cellular regulation is well recognized [38]. In agreement
tivity to explore the counting noise for purely diffusive withour results,inbiologicalsystems activetransportis
scenarios [1–11] and to analyze the speedup of recep- indeedemployedtomovelargerparticles(e.g.,vesiclesor
tor binding due to the topological coupling of one- and viruses)withintrinsicallysmallD [19,22–24,27,35]. As
three-dimensional diffusion for gene regulationin the fa- a result even during longer periods of detachment from
cilitateddiffusionmodel[12,37]. InthisLetterwefillthe motors these particles barely move [18, 22–24, 35]. The
apparentgapinthequantitativeassessmentofthecount- typical experimental values x 0.5 10µm [23–25, 35]
a
≃ −
ing noise experienced by biochemical receptors measur- for a 1 10nm in fact fulfill the requirements of our
≃ −
ingthelocalconcentrationofsignalingmoleculesorcom- modelforsignalfocusing. However,asdiscussedherealso
5
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6
Supplementary material
Signal focusing through active transport
Aljaˇz Godec1,2 and Ralf Metzler1,3
1Institute of Physics & Astronomy, University of Potsdam, D-14476 Potsdam-Golm, Germany
2National Institute of Chemistry, 1000 Ljubljana, Slovenia
3Department of Physics, Tampere University of Technology, FI-33101 Tampere, Finland
InthisSupplementalMaterialwesummarisethecalculationsandfullresultsforactivetransportcoupledtoreversible
binding to a receptor.
SPEED
In this part we neglect the binding to the receptor (see main text for argumentation) and invoke a probabilistic
interpretation of the model (i.e., concentrations probability densities). By Laplace transforming Eqs. (2) and (3)
→
in the main text in time and Fourier transforming in space, (r,t) (k,s) can be solved for c˜ and c˜ . The Laplace
a p
→
image of the mean squared displacement (MSD) for a particle starting at the origin in the passive phase is obtained
from r2(s) = 2k( c˜adΩ+c˜p)k=0 andinterchangingtheorderofdifferentiationwithrespecttokandintegration
h| |i −∇ |
with respect to Ω:
R
6D 1+ τp−1 +2v2s+τa−1+τp−1
r2(s) = s+τa−1 τp (s+τa−1)3 , (S1)
h| |i s+(cid:16) τp−1−(τa(cid:17)τp)−1/(s+τa−1)
which can be inverted exactly to give Eq. (4) in the main text. The equilibration time τ is defined implicitly by
i
r2(τ ) = L2, L being the linear dimension of the cell or its compartment. The transcendental equation for τ is
i i
h| |i
solved numerically.
SENSING PRECISION
The Fourier transformed linearized Eqs. (1) read
−iωδn˜(ω) = −τc−1δn˜(ω)+kon(1−hni)δc˜p(r0,ω)+koffhniβδF˜(ω)
iωδc˜(k,Ω,ω) = (iv(Ω) k+τ 1)δc˜(k,Ω,ω)+(4πτ ) 1δc˜(k,ω)
− a − · a− a p − p
−iωδc˜p(k,ω) = −(Dk2+τp−1)δc˜p(k,ω)+τa−1 δc˜a(k,Ω,ω)dΩ+iωδn˜(ω)e−ik·r0
Z
where we have used the constraint imposed by detailed balance δk /k δk /k = δF/(k T) when linearizing
on on off off B
−
the first of Eqs. (1) in the main text. Solving for δn˜(ω) we obtain
k n βδF˜(ω)
off
δn˜(ω)= h i , (S2)
τc−1−iω 1+kon[1−hni](2π)−3 dkΞ(k,ω)
whereβ =1/(k T)andwehavedefinedthecorre(cid:0)lationtimeτ (k c R k ) 1 fo(cid:1)rtwo-stateMarkovianswitching
B c on p off −
≡ h i−
[1] as well as the two auxiliary functions defined as
Ξ(k,ω) iω+Dk2+τ 1 (4πτ τ ) 1Ψ(k,ω) (S3)
≡− p− − p a −
π 2π dϕ
Ψ(k,ω) dθsin(θ) . (S4)
≡Z0 Z0 τp−1−i(ω−v(θ,ϕ)·k)
We are interested in the low-frequency limit and hence take Ψ(k,ω) Ψ(k,0). In this limit the integral over ϕ in
≃
Ψ(k,ω) can be evaluated as a contour integral along the unit circle via the method of residues, while the second one
is amenable directly leading to
4π
Ψ(k,0)= tan 1(vτ k), (S5)
− a
vk
7
where k k. Using Eq. (S5) we can obtain δn˜(ω) as a function of δF˜(ω) explicitly. The linear response function of
≡| |
the receptor occupancy (the coordinate) to a change in the free energy difference between bound and freely diffusing
species (the thermodynamically conjugate force) is then δn˜(ω)/δF˜(ω) and is related to the power spectrum of n via
the fluctuation-dissipation theorem S (ω) = 2/(βω)Im[δn˜(ω)/δF˜(ω)]. A change in concentration is equivalent to a
n
change in F as a consequence to a change in chemical potential, δc / c =βδF. Using this it can be shown [1] that
p p
h i
S (ω)=(β c )2 δn˜(ω)/δF˜(ω) 2S (ω) leading to
c p − n
h i | |
2 c 2 δF˜(ω)
p
S (ω)= h i Im . (S6)
c
−ωkBT "δn˜(ω)#
Weassumethatthereceptormeasuresandaveragestheconcentrationoveratimeτ muchlongerthananycorrelation
m
time. We are thus interested in the low frequency limit S (ω 0) leading to δc2 = S (ω 0)/τ and hence take
c → p c → m
Ξ(k,ω) Ξ(k,0). Finally, cutting the integral in the inverse Fourier transform k = π/a to correct for a finite
max
≈ | |
target size (and thereby avoid the UV divergence) we arrive at Eq. (5) in the main text with
1 a 2 tan 1(πx q/a) 1
Λ(x ,x )= 1+ 1 − a −dq. (S7)
a p
πx q − πx q/a
Z0 (cid:16) (cid:16) p (cid:17) h a i(cid:17)
The integral in Eq. (S7) can be evaluated exactly in the limits when the active excursions are either very short
x /a 1 or very long x /a 1 compared to the size of the target and gives Eqs. (6) and (7) in the main text.
a a
≪ ≫
Now we have to evaluate the equilibrium concentration in the passive phase in the general case, where signaling is
not necessarily transport controlled. In absence of a directional bias the concentration will be uniform in both the
active and passive phase. Hence we only need to solve a system of linear equations for n , c and c dΩ. It can
p a
h i h i h i
be shown that the steady state concentration of freely diffusing molecules c is given by
p
h i R
c V 1 k c V 1 k 2 c k 1/2
tot − off tot − off tot off
c = − + − + , (S8)
p
h i 2(1+τa/τp) − 2kon "(cid:18)2(1+τa/τp) − 2kon(cid:19) kon(1+τa/τp)#
where V is the volume of the cell or its domain. For k /(k c ) 0 we obtain c =c /(1+τ /τ ), where for
off on p p tot a p
h i → h i
convenience we have absorbedthe term 1/V in the total concentrationc V 1 c . Expressing the ratio τ /τ
tot − tot a p
in terms of x and x , τ /τ = D xa we observe that c and hence the r−elative→accuracy along with x and x in
p a a p |v|x2p h pi a p
fact depends on the factor D as well.
v
| |
[1] W. Bialek and S. Setayeshgar, Proc. Natl. Acad. Sci. USA102, 10040 (2005).