Effective Potential for Cellular Size Control

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by  David A. Kessler

Effective Potential for Cellular Size Control
David A. Kessler∗ and Stanislav Burov†
Physics Department, Bar-Ilan University, Ramat Gan 52900, Israel
Forvariousspeciesofbiologicalcells,experimentalobservationsindicatetheexistenceofuniversal
distributions of the cellular size, scaling relations between the cell-size moments and simple rules
for the cell-size control. We address a class of models for the control of cell division, and present
the steady state distributions. By introducing concepts such as effective force and potential, we
are able to address the appearance of scaling collapse of different distributions and the connection
between various moments of the cell-size. Our approach allows us to derive strict bounds which a
potential cell-size control scenario must meet in order to yield a steady state distribution. The so-
called“adder”modelforcell-sizecontrolexhibitstheweakestcontrolthatstillenablestheexistence
of stable size distribution, a fact that might explain the relative “popularity” of this scenario for
7
different cells.
1
0
2
Probably the most basic result of equilibrium statis- ble size distribution of the bacteria [15]. The protein

n tical physics is that knowledge of the potential energy number during the bacterial growth was successfully de-
a
functionandthetemperaturearesufficienttocompletely scribed by a model sharing similar principles [10].
J
describe a system’s long-time behavior [1]. The function Our goal in this manuscript is to address the con-
6
exp(−U(x)/kBT) is the only ingredient of the Boltz- nection between a given growth model and the emer-

] manndistribution, andaknowledgeoftheenergy(U)in gent steady state distribution. We will show that all
t
f anygivenstate(x)ofthesystem(foragiventemperature theabovementionedbehaviorsofrescalabledistributions,
o
T) determines the distribution of physical observables in connection between moments, i.e., Taylor’s law, and sta-
s
. equilibrium. The challenge of finding similar approaches bilitycriteria,canbedescribedintermssimilartoenergy
t
a foroutofequilibriumsystemshasfascinatedthescientific and temperature in statistical physics. We will derive
m community and many general advances in this field have an effective potential for cellular growth and show how
- been made. One of the most important examples of an the notion of effective temperature is different for vari-
d
out of equilibrium system is a living biological system. ous growth models. The relatively frequent appearance
n
o There are many differences between live biological sys- of the “adder” model will become clear when addressed
c temsandthephysicalsystemsthatareusuallyaddressed from the standpoint of the weakest form of the potential
[ in statistical mechanics. The tremendous complexity in that produces a stable distribution. Moreover, we show

1 biology makes the idea of existence of a single concept how the same criteria that gives rise to the adder model
v (like energy) that determines the observed rich behav- predicts the appearance of broad power-law tails for the
5 ior look like a non-realistic pipe dream. Nonetheless, the stable distribution.
2 presence of significant levels of noise together with the
Forthegrowthofabacteria,thereisagrowingnumber
7
known cases where simple statistical behavior emerges
1 of experimental studies that clearly show an exponential
out of the many underlying processes, gives hope for the
0 growth of the bacteria volume till its division. At divi-
. possibleapplicationofstatisticalphysicstospecificprop- sion, the bacteria splits into two parts, of equal size for
1
erties of living matter [2]. One such property is cellular
0 symmetric division and otherwise for asymmetric divi-
growth and division, which has been addressed in many
7 sion. Wewillrestrictourselvesinthisstudytosymmetric
1 recentexperimentalstudies[3–7]. Forproteinexpression division [18]. We define v to be the size of the bacteria
n
: inbacteriaaconnectionbetweenthefirstandsecondmo-
v afterthen’thdivision. Thesizeatwhichthebacteriawill
ments of cellular size was detected [8–10] (a relation also
i divide the next time is then 2v . In general, a growth
X n+1
known as Taylor’s law). The size of several unicellular
lawstatesthatthissize2v isdependentonthesizeof
n+1
r eukaryotes was observed to attain a stable distribution,
a thecellatthebeginningofthecycle, i.e. 2vn+1 =f(vn),
rescalablebythefirsttwomoments[3]. Similarbehavior
where f(·) is some specified function. The time differ-
wasobservedfortheproteinnumber[10]andcellsize[11]
ence between the n’th and (n+1)’th division is given
in E. coli. The distribution of inter-division times of C.
byt =τ[ln(f(v )/v )+η ]. Forsimplicity,thefluc-
n+1 n n n
crescentuswerealsoseentoexhibitascalingcollapse[12].
tuations in the growth rate 1/τ are neglected and the
Simplerulesthatgoverncellulargrowthwereverifiedex-
term η is the temporal noise accumulated through the
n
perimentally for several bacteria types [13–15]. For ex-
growth process. There are several competing scenarios
ample, in E. coli it was shown that the devision occurs
for the growth laws, i.e. f(·), that exist in the litera-
when the bacteria grows by a constant amount [13–15],
ture [14, 16, 19–21]. The models that have attracted the
a scenario termed as “adder” dynamics [16, 17]. This
most interest are: (i) The ”timer”: the cell grows for a
modelwasexploredtheoretically[16]and(underspecific
specificamountoftime(uptonoisyfluctuations). Inthis
simplifications)showntofitbeautifullytheobservedsta-
modelf(v )=θv , suchthattheaveragegrowthtimeis
n n
2
τln(θ). (ii) The“sizer”: the cell grows till a specific size
C. Heref(v )=C. (iii)The“adder”: aspecificamount 1.0 ○○○○ (�)
n ○ ○
of volume/mass is added through the growth process. ○
○
0.8
f(v ) = v +∆. The specific growth law for a specific ○
n n ○
species of bacteria must be inferred from experiment, al- ) ○
a 0.6 ○
tehraoOtuiugorhnbinatusvimiscnb+nme1orot=dnael21lwisvfaontyrhesetxhnapnet[lvhonaeb(rvsfiait(ootvuiconshn)at/saovtsfnikcc)e[+ml1l7aηs,pniz2]e(2.S]w.Mit)h ge(n1)- P( 000...024○♢□○○♢○□○♢○○□♢○○○♢□○○♢○□○♢○○□♢○○♢□○○♢○□○♢○○□○♢○○♢□○○♢○□○♢○○□♢○○♢○□○♢□♢□♢♢□♢□♢○□♢○○♢○□○○♢○□○♢○□○♢○○♢□○○♢○□○♢○○□♢○○○♢□○○♢○□○♢○○□♢○○♢□○○♢○□○♢○○□○♢○○♢
-3 -2 -1 0 1 2 3
It is more convenient to define a ≡ ln(v ) and then a
n n
Eq. (1) takes the simple form
1.0 ○○○○○○○○○○ (�)
a =a +g(a )+η , (2) ○
n+1 n n n ○
○
○
0.8 ○ ○
whereg(a )=ln[f(exp(a ))/2]−a . Duetotheexplicit ○
n n n ○
rdtcgaaeortelitontanawettirrinasmoo.slnciInarotbieftiseestbttrahtwaibhaeceleteefsneontrdraaiiaatsdqilntseurtstieaiiebczsnrauetmdi,ltoiibintontheniohenisfaagctsvsehwiutloelhffihrsesectoitazigfheebeneanirtvlneint.arty,aoImticntisroohiosntdorueewdrnffilieuaecrtmxihfetohabonritetbtoraihtbtnnoes- P(v) 0000....0246□♢♢♢♢♢♢□♢♢♢♢♢□♢♢♢♢□♢♢□♢♢○□○○♢○○♢○□○○♢○○○♢○○○□♢○○○♢○○○♢○○♢○□○○♢○○○♢○□○○♢○○○♢□○○♢○○□♢♢□♢□♢♢□♢♢○□○○♢○○♢○□○○♢○○○□♢○○○♢□○○♢○□○○♢○□○○♢○□○○♢□○○□♢○○□○♢○□○□○♢○□○□○♢○□○□♢○
map Eq. 2, where the function g(·) can be quite nonlin- 0 1 2 3 4 5 6
ear. The criteria for stability immediately rules out any v
growth laws that produce a g(·) that diverges too fast
1
afincnseIsdnucarsce)ardnciraiealLhebntnnbeaihedadtdneseln→ey,gnaaavaeaoffilvpp±oeaeLiwpasfrnc∞tgarrettatoenhed.ibo)xgqcelfileaueIameamvnsanndiiadtannaitiegetosiehaxerotdndorqtenibiw,semu[bc2ariaissurna3tftehtnta]iigt,iinolooic(wndzftnaaoaeailnstdwrnnwuh)cr/ioffiefitu∼sdoluhl[pncnr2tbimebeaδ3ec=o2nen,iwcufi,ot2ogone,f4c(rb(di]wriatat.tneeehnapdltTcmai)eeronhtnnofp+neeaevdetdd|siirηaadh.ntandnneiouto.vo|sEuonafiiaWrusoqsneels.wgtlshoaued((iuwigtn2nielfnhee)----. Prob.(v>x) 111100000----.15432 ○♢□○○♢□○♢○□○♢□○○♢□○○♢□○○♢□○○♢□○○♢□○○♢□○♢□♢□♢□♢□♢□♢□♢□♢□♢□♢□♢□♢□♢□♢□♢□♢□♢□♢□♢□□□□□□□□□□□□□□□□(□�□□)□□□
○ ♢
of such a description of the cellular division and growth 1 10 100 1000 104
process lies in the efficient mathematical tools that were x
derivedinordertodescribetheequilibriumbehaviorofa
system. The bacterial case is a nonequilibrium one, but
FIG.1: Distributionsofcellsizesv andtheappropriatevari-
thesizecontrolprocedureeventuallycreatesapopulation
able in log-space a = ln(v) for the adder model. Three dif-
with a stable size distribution. This stable distribution ferent noise strengths (cid:104)η2(cid:105) were used: (cid:104)η2(cid:105) = 0.1 (circles (cid:13)
willbeachievedbymeanssimilartohowtheequilibrium ), (cid:104)η2(cid:105)=0.1 (dimonds ♦ ) and (cid:104)η2(cid:105)=0.1 (squares (cid:3) ). The
distributionisobtainedforathermal(nonliving)system. parameter ∆ is 1 in all plots. Panel (a) displays the distri-
In recent study [25] we derived a general approach for butions of a, panel (b) displays the distributions of v. Thick
derivingacontinuousapproximationforSM’softheform linesare appropriateanalytic approximation by thevirtue of
Eqs. (4) and (6). Panel (c) displays the asymptotic power-
Eq. (2). The so-called second order approximation for
lawdecayofthev distributions. Thedecayisconsistentwith
Eq. (2) is a Langevin Equation with multiplicative noise
Eq. (7).
(cid:20) √ (cid:21)2
da = g(an)− 41∂2∂ga(a2nn) 1+12(cid:104)∂ηg∂2(aa(cid:105)nn) dn+ (cid:112)(cid:104)η2(cid:105) dB .
n 1+ 1∂g(an) 1+ 1∂g(an) n
2 ∂an 2 ∂an(3) term) and the brackets (cid:104)...(cid:105) represent the ensemble av-
erage value. Using the standard technique of deriving a
The generation number n is now treated as a continuous Fokker-Planckequationforthedistributionofa [24–26],
n
parameter, B is the Weiner process [23] (i.e., the noise thestabledistributionP(a)isoftheformofaBoltzmann
n
3
distribution energy should be ∼ ln(4)a. This explains the observed
Gaussian tail for negative a and exponential decay for
(cid:18) (cid:19)
H(a)
P(a)=N−1exp − (4) positive a. For the size itself, v, the adder mechanism
(cid:104)η2(cid:105)
then predicts a power-law asymptotic behavior,
where −(cid:16)1+ln(4)(cid:17)
P(v)∼v (cid:104)η2(cid:105) for v →∞. (7)
H(a)=−2(cid:90) g(a)da− 1g2(a)−(cid:104)η2(cid:105)ln(cid:18)(cid:12)(cid:12)(cid:12)1+ 1dg(a)(cid:12)(cid:12)(cid:12)(cid:19)
2 (cid:12) 2 da (cid:12) The power law behavior, which is a direct consequence
(5) of exponential growth and the “adder” scenario suggests
plays the role of effective physical potential energy. N = theappearanceofextremelylargecellsinthepopulation.
(cid:82)∞exp(cid:0)−H(a)/(cid:104)η2(cid:105)(cid:1) da is the normalization constant. When (cid:104)η2(cid:105) ≥ ln(4) the average size of the cell diverges,
∞
The analogy to physical situation is not perfect; the meaning that in such a case we will encounter bacteria
noise strength (cid:104)η2(cid:105) plays the role of the temperature that will grow for the whole time of the experiment (like
k T but it also appears in H(a), reflecting the presence the filamentous bacteria). Note that Ref. [16] approx-
b
of multiplicative noise [26, 27] in Eq. (3). Such a de- imated the “adder” with a log-normal distribution, in
scription of the effective potential energy allows the cri- contradiction to our Eq. (6), with its power-law tails.
teria for stability to be easily obtained: the potential
energy must be binding. This must be used with cau-
tion since too strong a divergence of H(a) will produce 1.0 ○♢○x♢○x♢x○♢x○♢x (�)
○♢x○♢x
a situation where no stable distribution for a exists, as
○♢x ○♢x
pmroedveiol,uwslyhemreengt(iao)n=edl.n(Aθ/s2i)mipsleaecxoanmstpanlet.isTthheef“utnimcteiorn” v〉) 0.8 ○♢x ○♢x○♢x
exp(cid:0)−H(a)/(cid:104)η2(cid:105)(cid:1) is non-normalizable and a stable dis- v/〈 0.6○♢x ○♢x○♢x
tribution does not exist. Eq. (3) for such a case displays P( 0.4○♢x ○♢x○♢x
○♢x
a behavior of a particle pushed in a specific direction by ○♢x
○♢x ○♢x
aesiqzcueoinlciosbtnraitunrmotlf.oscTrechnee,arnaioosniwt-uvaaasltiiadolinrteyatohdfaytthnceoat“net’ditmianetrtR”aieanfs.pa[1op6soi,ts1iso7inb].alel 00..02○♢○x♢○x♢x ○♢x○♢x○♢x○♢x○♢x○♢x○♢x○♢x○♢x○♢x○♢x○♢x○♢x○♢x○♢x○♢x○♢x○♢x○♢x○♢x○♢x○♢x○♢x○♢x○♢x○♢x○♢x○♢x○♢x○♢x○♢x○♢x○♢x○♢x○♢x○♢x○♢
0 1 2 3 4 5 6
We continue the exploration of possible scenarios. For
v/〈v〉
the “sizer”, we have g(a) = ln(C/2)−a and so H(a) =
(ln(C/2)−a)2/2,accordingtoEq.(5). Thisisacaseofa 10 ○
(�)
quadratic effective energy around the point a=ln(C/2).
○
Our approximation for such case produces the Gaussian
5
distribution P(a) ∼ exp(−(ln(C/2)−a)2/2(cid:104)η2(cid:105)), which 〉) ○
2
is exact for the linear map [16, 25, 28]. The appropriate η ○
〈
distribution of the size v is then log-normal, i.e., P(v)∼ ( ○
(1/v)exp(−[ln(2a/C)]2/2(cid:104)η2(cid:105)). mp 2 ○
lnT[(ehxep(tah)ir+d∆s)c/e2n]a−rioa iiss htehree n“oand-dlienre”arw.hTerhee egff(aec)tiv=e 1 ○♢ ○♢ ○♢ ○♢ ♢ ♢ ♢ ♢ ♢ ♢
energy H(a) for this case is
0.0 0.5 1.0 1.5 2.0
H(a)=ln2(2ea/∆)+2Li (−ea/∆)− 1ln2(cid:18)ea/∆+1(cid:19) p
2 2 2ea/∆
(cid:18)2ea/∆+1(cid:19)
−(cid:104)η2(cid:105)ln FIG. 2: Stable behavior of the cell size in the adder model.
ea/∆+1
Panel (a): distribution ofthe cell size collapsewhen normal-
(6)
izedbythemean. Thesymbolspresentdifferent∆s: ∆=0.1
(circles (cid:13) ), ∆ = 1.9 (dimonds ♦ and ∆ = 5.0 (crosses ×).
Here, Li (x) is the polylogarithm function [29, 30]. The
2 Thethicklineistheanalyticsolution. Panel(b)presentsthe
asymptotic behavior of H(a) [31] is H(a) ∼ a2/2 as behaviorofthemomentsofthecellsizefor(cid:104)η2(cid:105)=0.5(circles
a → −∞ and H(a) ∼ ln(2)a as a → ∞. This result (cid:13)) and (cid:104)η2(cid:105)=0.1 (squares (cid:3)), ∆ equals 1.0 for both cases.
can be simply explained by looking at the form of g(a). Thicklinesrepresenttheanalyticsolutionfortheaddermodel
Considering g(a) as an effective force and using the ef- and dashed lines are solutions of the linearized adder model.
fectiverelationbetweenforceandpotential, asdescribed
by Eq. (5), we obtain: (i) for large negative a the force The comparison between the distribution P(a), as
is∼−a,implyingthatthepotentialenergy∼1/2a2,(ii) given in Eq. (4), for the three model g(·)’s and direct
for large and positive a as g(a) = −ln(2) the potential simulation of the SM, Eq. (2), is shown in Fig. 1. Very
4
good agreement is observed not only at the center of the linear at most. Basically it means that the size control,
distribution but also for the tail behavior. g(a), isboundedbetweensomeconstantvalueandlinear
The effective energy H(a) in Eq. (6) can be written as growth as a → ±∞. Specifically, the adder scenario ful-
a function of two variables: exp(a)/∆ = v/∆ and (cid:104)η2(cid:105). fills the first and second conditions. What is important
Thisfactissufficienttoestablishaseparationofvariables to notice is that for large values of v the adder scenario
forthemomentsofv,i.e.,thecellsize. Eachmomenthas is the least restrictive, as g(a) → −ln(2) as a → ∞.
the form (cid:104)vp(cid:105) = m ((cid:104)η2(cid:105))∆p, where is m (...) is some What is meant by “least restrictive” is that the minimal
p p
functionthatdependsonthemomentpowerp. Moreover, effective force is applied in order to stabilize the bacte-
any rescaling of v by A((cid:104)η2(cid:105))∆ will produce a distribu- ria size. When treating small sizes, the adder scenario
tion that is independent of ∆ and a “distribution col- is much more restrictive and a maximal effective force
lapse”willoccur(forfixed(cid:104)η2(cid:105)),asshowninFig.2. The is applied for stabilization. From this discussion it be-
power-law dependence of the moments on ∆ and separa- comes clear that the least restrictive scenario is when a
tion of variables can be viewed as specific manifestations minimaleffectiveforceisappliedforbothlargeandsmall
ofTaylor’slaw,i.e.,(cid:104)v2(cid:105)∝(cid:104)v(cid:105)2[32–34]. Thisphenomena cell sizes v. For large sizes, as we noted, it is an adder-
was recorded for many physical/biological/ecological sit- like scenario. For small sizes, the minimal force (g(a))
uations,andspecificallywasobservedforproteinnumber is a positive constant, i.e., a timer scenario. It is then
expression in E. coli [8–10]. a mixed scenario of timer for small sizes and adder for
While we utilized a continuum approximation for the large sizes that is a minimal scenario capable of stabi-
SM, a common practice is a linearization of the map lizing cell size. While we can’t claim that the cell must
around the fixed point. Specifically, for the “adder” sce- prefer such a mechanism in order to minimize the effec-
nario, the linearization of g(a) around ln(∆) produces tive energy invested in controlling the size, it is still very
a linear map a = a −1/2[a −ln(∆)]+η . This encouraging that exactly such a “mixer” mechanism was
n+1 n n n
map produces a quadratic effective energy with a mini- very recently spotted for C. crescentus [22].
mumata=ln(∆). Thelinearizedaddereffectiveenergy Anadditionalconstraintthatmustbesatisfiedbyany
is shifted with respect to the adder scenario. This fact growthscenarioisthedistributioncollapseduetorescal-
canbeobservedfromthelocationofmaximumofvP(v), ing. From the form of the effective energy in Eq. (5)
whichislocatedatv =∆forthelinearizedadderandat and the growth scenario g(a) we can conclude that for
v ∼(1+γ)∆(whereγ =6(cid:104)η2(cid:105)/(27+(cid:104)η2(cid:105)))fortheadder any growth scenario that can be written as some func-
case. The presence of non-linearity and multiplicative tion of a−f(λ ,λ ,...,λ ) (λ are some parameters of
1 2 n i
noise is what is responsible for this shift. Not only the the scenario) a distribution collapse can be observed. If
location of the minimum but also the shape of the effec- the different distributions were created by changing only
tive energy is quite different, with a linear divergence of the parameters λ a collapse of the distributions will oc-
i
theadderasopposedtoaquadraticdivergenceofthelin- cur if the size v is normalized by (cid:104)v(cid:105). This effect is dic-
earizedadder. Thisshowsitselfquitestronglyinthetails tated by the form of H(a) in Eq. (5) and the fact that
and higher moments of the distribution, which strongly (cid:104)v(cid:105) ∝ exp[f(λ ,λ ,...,λ )]. This effect can be used as
1 2 n
deviatefromthelinearizedversionsasthemomentpower an examination of which parameters were changed un-
grows (see Fig. 2). dervariousexperimentalconditions,whennon-collapsing
We have so far addressed the properties of three spe- distributions will have changes to parameters that are
cific size control scenarios using our continuum approx- not in the set λ1,...,λn. For example, in the case of
imation of the stochastic map, characterizing the prop- the adder scenario the collapse will occur only if ∆ is
erties of the stable distributions and emergent features changed, whilechangingthenoisestrength(cid:104)η2(cid:105)willpro-
like the power-law behavior and discrepancies between duceadistributionthatdoesnot“collapse”underrescal-
the full and linearized size-control model. An additional ing.
consequence of our formalism is that the form of the ef- An intriguing point is why in living systems such sim-
fective energy in Eq. (5) imposes bounds that must be ple scenarios for cell-size control appear. While it is
satisfied by any possible size control scenario. First, the hard to address this question from a molecular perspec-
effective force g(·) must be negative for large positive tive, the treatment of cell-size control as a map permits
values of ln(v) and positive for sufficiently large negative quite general statements. Two important points must
valuesofln(v). Ingeneralitmeansthattheeffectiveforce always be satisfied, (a) Existence of a fixed point for
mustberestoring,i.e. particleconnectedtoaspringthat the map and (b) Stability with respect to noise fluctu-
perfectly describes the linearized adder. Second, even ations. Any molecular cell-size control mechanism must
when the first condition is fulfilled, due to the presence eventually satisfy those restrictions. Our previously de-
of −g2(a)/2 in the effective energy the behavior of g(a) veloped approximation of stochastic maps allows us to
in the limits a → ±∞ is restricted. Since the size dis- describe these restrictions in the language of physics us-
(cid:82)
tribution is assumed to be stable, the term −2 g(a)da ing such terms as potential and force. General bounds
must be dominant, restricting the growth of g(a) to be are then naturally obtained in terms of properties of the
5
force/potential. [16] A. Amir, Phys. Rev. Lett. 112, 208102 (2014).
[17] S.JunandS.Taheri-Araghi,TrendsinMicrobiology23,
4 (2015).
An additional advantage of the ideas of effective force [18] A. Maranatan and A. Amir, Phys. Rev. E 94, 012405
and energy for cellular growth and devision presented (2016).
herein is the the possibility of treating static and time- [19] S.Cooper,Bacterial Growth and Division: Biochemistry
dependent perturbations. This will allow for the theory and Regulation of Prokaryotic and Eukaryotic Division
Cycles (Elsevier, New York, 1991).
to deal with with experiments that take the “top down”
[20] A. L. Koch, Bacterial Growth and Form (Springer,
approach and try to affect (and study) the single cell
Berlin, 2001).
behavior as a whole by application of external macro-
[21] W.L.VoornandA.J.H.Koppes,Arch.Microbiol.169,
scopic perturbations, e.g., chemical perturbation spec- 43 (1997).
troscopy [35]. [22] S. Banerjee et al., bioRxiv, 047589.
[23] C. Gardiner, Stochastic Methods: A Handbook for the
NaturalandSocialSciences (Springer-Verlag,BerlinHei-
delberg, 2009).
[24] H. Risken, The Fokker-Planck Equation: Methods of
∗ [email protected] Solution and Applications (Springer, Berlin Heidelberg,
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