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OPEN Kinetic constraints on self-assembly into closed supramolecular structures

Received: 11 May 2017 Accepted: 4 August 2017 Published: xx xx xxxx

Thomas C. T. Michaels1,2, Mathias M. J. Bellaiche1,3, Michael F. Hagan4 & Tuomas P. J. Knowles1,5
Many biological and synthetic systems exploit self-assembly to generate highly intricate closed supramolecular architectures, ranging from self-assembling cages to viral capsids. The fundamental design principles that control the structural determinants of the resulting assemblies are increasingly well-understood, but much less is known about the kinetics of such assembly phenomena and it remains a key challenge to elucidate how these systems can be engineered to assemble in an efficient manner and avoid kinetic trapping. We show here that simple scaling laws emerge from a set of kinetic equations describing the self-assembly of identical building blocks into closed supramolecular structures and that this scaling behavior provides general rules that determine efficient assembly in these systems. Using this framework, we uncover the existence of a narrow range of parameter space that supports efficient self-assembly and reveal that nature capitalizes on this behavior to direct the reliable assembly of viral capsids on biologically relevant timescales.

The spontaneous formation of nanoscale materials with specific chemical and physical characteristics from basic molecular building blocks is a key process for the functioning of living systems and provides a bottom-up strategy for constructing novel nanomaterials for various applications1. A particularly important class of such molecular self-assembly processes is the formation of closed supramolecular structures, with examples including clathrin assemblies2, self-assembling cages3,4, micellar-like structures5, small polyhedra6–8 or icosahedral viral capsids9–11. Many assembly processes of this type underlie key events in normal biology12, but are also implicated in the onset of diseases of humans, animals and plants. Moreover, the construction of such molecular topologies offers great potential as biomimetic nanocontainers for encapsulation, delivery and release of small molecules.
Elegant physical principles have emerged that determine the geometric and equilibrium constraints governing the shapes of the resulting assembly structures in these systems, motivating the question of whether or not analogous principles can be defined for their assembly kinetics. Probing molecular reaction mechanisms in complex systems represents a fundamental challenge through the Chemical Sciences; in this context, chemical kinetics has proven to be an extremely effective tool for testing mechanistic hypothesis in areas ranging from small molecule chemistry to enzyme kinetics. Recent advances have extended the applicability of this chemical kinetics approach to the study of filamentous protein assembly phenomena, such as amyloid formation13,14, providing fundamental insights into the nature of the microscopic steps in the aggregation process15–18. These advances have been made possible by the discovery of integrated rate laws that allow relating experimental measurements to the underlying microscopic mechanisms and hence studying the self-assembly into open-ended fibrillar structures at a highly detailed level15–18. It has however remained challenging to exploit the full power of the chemical kinetics approach beyond fibril formation to probe the molecular-level mechanisms of the more complex phenomenon of self-assembly into closed supramolecular structures, a difficultly originating in large part from the absence of integrated rate laws describing such processes. Here, we make a step forward in this direction by deriving a closed-form solution to a set of rate equations describing the assembly kinetics of molecular building blocks into closed target structures19,20, and show how the availability of this integrated rate law uncovers, from a kinetic

1Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge, CB2 1EW, UK. 2Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, 02138, USA. 3Laboratory of Chemical Physics, National Institute of Digestive and Diabetes and Kidney Diseases, National Institutes of Health, Bethesda, MD, 20892, USA. 4Department of Physics, Brandeis University, Waltham, MA, 02454, USA. 5Cavendish Laboratory, Department of Physics, University of Cambridge, J J Thomson Avenue, Cambridge, CB3 1HE, United Kingdom. Correspondence and requests for materials should be addressed to T.P.J.K. (email: tpjk2@cam.ac.uk)

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Figure 1. (a) Schematic representation of assembly line model: subunits nucleate first and then proceed downhill through elongation reactions to the final structure. Structures in the scheme exemplify assembly with nVbMlciu −=reu2 )3ssa−wan1nidktdh+dN m=e f=i (5n0. 3i)6t 0= i×o.  n(3 1bs.08)o5,CfM5co.4h−m,a16prs.aa−4cr1,ti8aseo.nr2indsatmonicfd(nt0i1u)m0 =m.e8 se1μr0timM cμaaxM.laDs.no(adlctua–tt1eif/o2r).onGCmtlaoo1l9bcE.uaq(lldsaf)it(ti1Hso)nouafmpnvadaarnr(ai2moP)uae(pstdeivalrlissorh:umeNsda k=vbiin lr9aeu0cts,kicnw)scw.i =t(hict 3)hm,HEk(0neq p)=. =a( 91ti )0 t×i.(s4s 10oB0,l6i0d.41, 0.53, 0.72, 0.74, and 0.80 μM. Data from21. (e) Brome Mosaic Virus with m(0) = 6.2, 11.1 and 14.0 μM. Data from22. (f) Extracted elongation and nucleation rate constants for all viral systems considered. Note that all experimental data analyzed in this work were obtained using purified proteins. Viral images reproduced from23 with permission.

analysis of experimental data, general dynamic constraints on the microscopic rate constants that control efficient supramolecular self-assembly in such systems.

Results and Discussion Fundamental kinetic equations.  The self-assembly of molecular building blocks into closed target struc-
tures may be captured by the following set of kinetic equations for the concentration f(t, j) of intermediates of size j, known as the assembly line model (Fig. 1(a))19,20:

∂f (t, j) ∂t

=

k+m(t)f (t, j − 1) − k+m(t)f (t, j) + knm(t)ncδj,nc

∂f (t, N ) ∂t

=

k+m(t)f (t, N − 1),

(1)

where N is the number of subunits in the target structure and m(t) is the concentration of free subunits in solution, as determined by conservation of the total subunit concentration

dm(t) = − d ∑N jf (t, j).
dt dt j=nc

(2)

The terms on the first line of Eq. (1) describe the growth of assembly intermediates through the addition of idntscnhcoieesdenosnisucovatrsicardiiintfaouiotct,arekaclmnlabs.nnuaaTtucbbihkucoeulnntetsohiu,ototssffhruwsetegiheizhpteeshtausirormnbaafumtaacensllealictettsohesssrnt,eicssgnsautircazlcocnenhwotourttkfcrhh+etla-ehs.ctapTeottohhsimonmeenpditaresetlrtclhteemooesnnotttkchrsnietyenman)tbr;te(elritreana)mcntatitcesoeidsrodnemenimsacecotabdrernliidybfaberetioreensmsotatehwfsrtsetmihuhtihemenediinsetniiiudazattceeletlroneja( au<ncbtctcie inloioesnncanteathsigrotoeelefinpagunnisncabtanseslletupoda.b2g,bF4uoilniaeunnsstiaatthnqlshleudyew,sasqiitnmpthuthoeiiptcnryllkaatetatlssoye-tt equation of (1) describes the end step of the assembly line as the closure of an intermediate of size N − 1 into the final structure. Note that in the limit of infinite N, Eq. (1) recover the kinetic equations commonly used to describe filamentous protein assembly processes14–16. Note that, in Eq. (1) we have assumed size-independent rate constants; this assumption was primarily made for minimizing the number of model parameters to avoid over-fitting in the analysis of kinetic data, but our framework can be extended straightforwardly to take this effect into account. Moreover, Eq. (1) are deterministic and neglect therefore the potential effect of statistical number fluctuations. Such fluctuations are often negligible in reactions in bulk, but can become dominant in reactions

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under volume confinement25. We also note that Eq. (1) assume that assembly results in a single closed capsid geometry. Application to reactions that yield multiple capsid morphologies, such as the CCMV systems studied in refs26,27, require a more complex master equation.

Integrated rate law for assembly kinetics.  To obtain an integrated rate law to Eqs (1) and (2), we make

use of the perturbative renormalization group (RG)28, a general mathematical technique for constructing approx-

imative solutions to nonlinear differential equations. In our case, the applicability of this method relies on the

soubbsuernvitatcioonnctehnattratthieond.iImneonrsdieorntloestsakraetaiodvεa≡ntakgnemfo(0rm)nca−ll2y/ko+f

 the

1 is a small parameter, where m(0) is the initial smallness of ε, it is convenient to rewrite Eq. (1)

in dimensionless form

∂φ(τ, j) ∂τ

=

µ(τ)φ(τ, j − 1) − µ(τ)φ(τ, j)

+εµ(τ )nc δj ,nc,

(3)

wsohmeereaτlge=brka+, mth(e0)fot ,llµo(wτi)ng=remsu(τlt)f/omr (t0h)eatnimd eφ-(vτa,ryj)in=g

f (τ, j)/m(0). Solving Eq. subunit concentration:

(3)

perturbatively

yields,

after

µ(τ) = ρ − εNρncτ

∑+ερnc−1Ω0

−

e−ρτ

N

−nc− k=0

1
Ωk

ρkτ k!

k



+

,

(4)

where ρ = 1 and the emergence of

Ωa dk i=ver(gNen−t tnerc m−(

k)(N − −εNτ)

antcla−terkt+im1es),/2w.hWichhilpereEvqe.n(t4s)tihsisaclicnueraartiezefodresahrolyr-ttitmimeesso, lwuteioonbsferorvme

being valid over the full time course of the reaction. At a fundamental level, this divergence emerges due to our

ignorance about the system’s behavior in the future; in fact, while the information about the initial concentration

of subunits is sufficient for describing the system dynamics over short timescales, at later times the lack of infor-

mation about the way in which ρ varies with changing timescale is what causes Eq. (4) to depart from the true

solution. Perturbative RG provides a systematic method for dealing with this undesired divergence and hence

obtain a global approximation valid for the duration of the whole reaction. Note that this procedure mirrors very

closely the conventional RG approaches of quantum field theory and condensed matter physics. In these theories,

we are interested in describing how a certain quantity of interest, such as the charge or the mass of an electron, is

renormalized as we vary the observation scale (e.g. momentum or energy scale in quantum field theory). The

missing information about the large-scale (e.g. high-energy) behavior of the system is packed into so-called coun-

ter terms, which are constructed in order to cancel the divergencies in the theory. In our case, the analogous

quantity to the electron charge or mass of quantum field theory is the initial concentration of monomers and the

RG procedure should yield renormalized values for this quantity at different time scales. Following the conven-

tional work-flow of RG, we start by introducing an arbitrary time scale σ which we will vary between the initial

time 0 and the observation time τ and then allow for a σ-dependence of the initial subunit concentration by

writing ρ = ρ(σ) + εδρ(σ), where ρ(σ) is the renormalized subunit concentration (at scale σ) and δρ(σ) is a

counter term. The counter term δρ(σ) = Nρncσ removes the divergent term in Eq. (4) and so we arrive at the

following renormalized expansion

µ(τ) = ρ(σ) − εNρnc(τ − σ) + ,

(5)

where  stands for regular terms. As a next step in the RG framework, we require ∂µ/∂σ = 0 since σ is arbitrary. Doing so, we arrive at the following RG equation

∂ρ ∂σ

=

− εNρ(σ )nc .

By solving Eq. (6) and substituting in Eq. (5) as σ → τ we obtain the uniformly valid solutions

(6)

∑µ(τ)

=

ρ(τ)

+

ερ(τ )nc −1Ω0

−

e−ρ(τ

)τ

N−nc

−1
Ωk

k=0

ρ(τ)k k!

τ

k



,

(7)

∑φ(τ,

j)

=

ερ(τ )nc −11

−

e−ρ(τ)τ

j−nc k=0

ρ(τ)k k!

τ

k

.

(8)

Finally, using conservation of total subunit concentration and transforming back to real time t we arrive at the following integrated rate law for the concentration of closed target structures:

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f (t, N ) = m(0) − ρ(t) − (N − nc)knρ(t)nc−1 N k+

∑×1

−

e−k

+ρ(t)t

N

−nc− k=0

1

(N − (N

nc −

− k) nc)

[k+ρ(t)t]k k!

,

(9)

where

ρ(t)

=

[1

+

N (nc

−

m(0) 1)knm(0)nc

−1t]1/(nc

−1)

.

(10)

This solution shows overall good agreement with the numerical evaluation of Eqs (1) and (2) (Fig. 1(b) and see Supplementary Material for a discussion on the accuracy of Eq. (9) as a function of ε). Moreover, we note that within the first-order RG approximation discussed here the kinetic trace for capsid formation is systematically underestimated by the analytical solution. This is because the function ρ(t) obtained by solving the first-order RG equation decays faster than the true solution. These errors can be reduced by applying the RG method to higher orders in ε.

General characteristics of assembly kinetics.  Using the integrated rate law, Eq. (9), we are now in the

position to derive, from first principles, a number of relationships characterizing the time course of the assembly

reaction. According to Eq. (9), the time evolution of the concentration of target structures demonstrates the char-

acteristic sigmoidal shape defined by an initial lag phase followed by a phase of rapid growth and final asymptotic

approach to the plateau20. A defining feature of the early time behaviour is at which the growth rate r = df(t, N)/dt is maximal. Solving the equation inflection point as

the presence of a point of inflection dr/dt|tmax = 0 yields the position of

ttmhaex

tmax

=

N − nc k+m(0)

.

(11)

The time of inflection is determined completely by the characteristic

physical interpretation of is consistent with the idea

Eq. (11) that the

is that of lag phase

tohfetthiemreearecqtiuoinrecdorforersNpo −n dnsc

eelloonnggaattiioonntsitmepessctaoleo(ckc+umr. (T0h)i)s−1r.eTsuhlet to a waiting period during which the

assembly line is set up and all intermediate states are populated20.

The maximal growth rate rmax = df (t, N )/dt|tmax is computed from Eq. (9) as

rmax

=

k nm(0)nc (N (N −

− nc)N−nc nc)!

e−(N−nc).

(12)

opfreodNbisocettreiovtnihnaogtftrhEmeaqx.mi(s1ign2ii)vmeisnaltbhnyeutmehmebpeerrrgoNedn u−cce tnocofof tfahepeloornawtgeearot-ifloarnwatsest-eclpaimlsinrietgiqnougfirntehudecltmeoaactixooimnmpsatlleegpteraotnhwdethtahsresae Ptmeobiwslsiytohsntiiranunicttpiuarlroesb.uaAbbuiklnietiyyt

γcoot =hne cnrec,natrthreaeatscioroinftisrcmcaialexnn~ucceml2e9u,(300s,)sγsi.czaBelecincaagnulbsaeewdtsheeetmesrcmearliginneegidnefxtrhpoemonctoehnnettesγxlotspooelfeoslyuf apdreloapmge-nolodlegscpoulnloattrhoaefsrnsmeaamtxuvbrselymoaf(s0tha).egTnehnuuecsrl,eaaalstpiiornonmpsetarentpyy,

that connects macroscopic data with the physical nature of the underlying microscopic processes through the

value of the scaling exponent.

Equation (9) implies that the median given by t1/2 = 2tmax + tnuc, where

assembly

time

t1/2,

defined

by the

condition

f (t1/2,

N)

=

m(0)/(2N ),

is

tnuc

=

N (nc

2nc − 1 − 1)knm(0)nc−1

(13)

is the time needed to consume half the free subunits after the assembly line is set up, assuming that each nuclea-

tion event leads to the target structure through the eventual consumption of N subunits (see Supplementary

Intmnuafcxo,lretmahteaettaiiomsnue)f.fniTcehiceiensstrseaasmruyolttuosnhftooorwfmsintahteqartumtahseeid-itsa1t/te2eaisdstyghisavtteamnteaaotsufariensutienmrtmootefhdtewiafotiendsa,ilsastnitnrducttchtcueornoetttrhhiebrruofturigoohnmst,htoneunct e+heo trcmihgaxai,nintahtreientagicmftrieoonmtos

of the assembly line. The former contribution to the assembly line, while the latter is governed by

nt1u/2cldeeaptieonndesvoennltys.oCnrtuhceiaelflfyi,ctihene cryeloatfitvheeimelponorgtaatniocne

reactions in of these two

contributions to the median assembly which measures the ratio of the rates of

tniumceleiastidoentearnmdienleodngbaytitohne–pnaartaumraelltyeremε =ergkensmfr(o0m)nco−u2r/kt+h.eTorheitsicqaulafnratmitye–-

work as the key parameter controlling the assembly kinetics. In general, large values of ε correspond to a kinetic

trap, whereby subunits are significantly depleted by nucleating too quickly, leaving less material to complete the

assembly of target structures. By contrast, when ε is small, few nuclei are formed and the assembly yield is low for

relevant time this criterion

scales. can be

The crossover between these formulated as a condition on

two regimes occurs the parameter ε as:

when

tnuc = 2tmax.

Using

the

results

above,

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εc

=

2nc−1 − 1 2N(N − nc)(nc −

1) .

(14)

AccWorhdeinngεto >t hεics,

the system is susceptible to criterion, successful assembly

kinetic traps, is the result of

awdheelriceaatsewbahleanncεe <be εtwc teheen

assembly is inefficient. the necessity of forming

appreciable amounts of target structures and the danger of being kinetically trapped. Controlling the relative

importance of nucleation and elongation processes provides therefore a high degree of intrinsic regulation of

self-assembly20. follows intuition

We note because

ltahrgaterεtcadregcertesatrsuesctwuritehs

increasing size N of the target impose stronger constraints on

geometry as N−2. the time available

This behavior for nucleation,

ftonurc ~la r1g/eNr,

while N.

the

time

required

for

producing

the

quasi-steady

state

assembly

line,

tmax ~ N,

is

inevitably

longer

Kinetic analysis of experimental data.  Through the analysis of experimental kinetic data, we now

demonstrate that the theoretical framework provided by Eq. (9) is capable of describing macroscopic features

of supra-molecular self-assembly into closed topologies in terms of microscopic rate constants. We took a

representative example and considered kinetic data of the formation of icosahedral viral capsids. Since the

current version of our theory only considers empty capsid assembly, we limit our comparison to in vitro exper-

iments on the assembly of purified capsid proteins; i.e., the systems do not include viral genomes, other viral

proteins, or host factors. Previous studies modeling viral capsid assembly kinetics using master equa-

tions19,20,31–36, continuum models37–39 or molecular dynamics simulations40–49 have led to important insights

into the system characteristics, yet it remains a key challenge to elucidate the general physical principles under-

lying capsid assembly. First, we consider the assembly kinetics of Human Hepatitis B Virus (HBV)19, a repre-

sentative icosahedral virus comprised predominantly of N = 120 subunits. Figure 1(c) shows the time evolution

of HBV capsid concentration, as monitored by light scattering intensity, fit globally to the integrated rate law

Essttqara.nt(te9ss)otwfhkie+thc=ofi3nx.se3id2stne±nc =t0a. 31g5r(ea×esmd1ee0tne5trMmbe−itn1wese−de1fnaronEmdqk.t(nh9=e)

scaling of maximal growth rate, Fig. 2(b)), yielding rate con1.6 ± 0.9 × 106 M−2 s−1. The global nature of the fit demonand the full time courses observed in the experiment over a

wide range of initial subunit concentrations, including the characteristic sigmoidal shape of kinetic traces. We

note that the entire data set could be fitted to Eq. (9) using just two global rate constants and one

concentration-dependent plateau parameter for each kinetic curve that accounts for the constant of propor-

tionality between the measured light scattering signal and the capsid concentration. In the SI, we provide also

fitting to HBV assembly data under reducing conditions at 37 °C and pH 7.5 from ref.50. The fits to these higher

temperature data, however, are less accurate, which could arise due to late stage intermediates52 and protein

interconversion between assembly-active and assembly-inactive conformations53,54. Next, we consider kinetic

data for the formation of Human Papillomavirus (HPV, Fig. 1(d))21 and Brome Mosaic Virus (BMV, Fig. 1(e))

kFckan+up r==stihd 89es.r.290m2 .±±oUr s0e0i,.n.58fgr ×o×nm c1 =10t0h 626e,MMNan−− =a21 l s7ys−−2s1i1s((,HBokMfPntVh=V)e)a6ean.xr4dpee±nargci =am0i.n 4e3n,a×Ntbal l=e1k0t 9io2n0deM(teBis−cMc1driVsab−t)ea1,,tg(whlHoeebPfcuaaVllnlf)itatiaslmsnooedfdcekoix+rupere=csrteliym5o.v6feentr±htiaefyl1ad.ts0hastee×amstcob1al0Elyi5nqrM.ge(ap9−cr)t1eiwodsni−itcs1h-,.

tions that have resulted from our analytical treatment of the master equation (1) for the three virus systems

discussed here. Figure 2(a) tial subunit concentration

shows for the

athdroeuebslyesltoegmasricthomnsiicdperloetdoifntFhiegm. 1eatosugreethdeirnwfleitchtiothnetipmreedsi,cttmeadx,

against scaling

inilaw,

tkminaxet~icmp(r0o)f−il1e. sTchloe ssectaotttehreinintihtiealdpaotainftoorf tthhee

inflection time is due to increased experimental noise in the reaction. Moreover, Fig. 2(b) illustrates how the relevant value

for the rate as

reaction order for a function of total

stuhbeunnuictlceoatnicoennsttreapti,onnc.,

can be determined from the analysis Through the analysis of the maximal

of the maximal growth growth rate, it is there-

fore possible to fix the value of set to 3 as reported previously

innc

tnheecleistseararytufroer22f;ittthinisgwkainsedtoicnterabceecsa.uWseetnhoetceotrhraetspthoenvdainlugedoaftansceftohr aBsMtoVo

was few

points for confident fitting. We also note that similar scaling laws have been previously obtained approximately

by assuming an ad hoc separation between nucleation and growth processes39,55.

The availability of microscopic rate constants enabled by the present analysis allows mechanistic comparisons

to be made between the assembly of different virus capsid systems. Interestingly, while the absolute values of the

rate constants obtained from the fitting of experimental data vary over several orders of magnitude (Fig. 1(f)), we

observe that the parameter ε takes similar values across the different data sets: ε = 5.0 ± 0.4 × 10−5

(m(0) = 10 µM) for HBV, ε = 8.0 ± 0.7 × 10−5 (m(0) = 1 µM) for HPV and ε = 1.7 ± 0.3 × 10−5

d(tmiios(nt0isn)cf=otrv1iεrc0a:µl5s.My5s)t×efom1rs0B−sMt5u(VdHi.eBMdVoin)r,et1oh.v0iser×w, oth1r0ke−sae4rve(HaclhuPaeVrsa)fcaatlnelrdiinz9et.dh5eb×ysaa1m0si−em5oi(rlBdareMrbVaolf)a.mnTcahegionsfiittluhludesertreaalsattetihsveehtorhaweteotsrheoetfiacepalolpnpagrraeetndioitclny-

and nucleation to achieve successful assembly. By contrast, for filamentous protein self-assembly14–16 the

tlohneigr-btiimoleogavicearlafguenlcetniognthtoofbaeglgornegg,atthesey〈Ls〉hboeucldomhaevse〈Llo〉w~m1e/asεucr.eAdsεlisnoeaasr

systems such as actin51 to maximize efficiency.

are required by This prediction

is in agreement with what is observed in Fig. 2(c).

Conclusions
In conclusion, although it is of both fundamental and practical interest to identify and characterize the kinetic constraints governing supra-molecular self-assembly into closed target structures, this understanding has proved challenging to achieve in practice. Here, we have demonstrated how the availability of integrated rate laws to

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Figure 2.  Scaling m(0) as predicted

bbyehEaqv.i(o1r1o)ffvoirrvaal rciaopussidviarsaslesmysbtleympsrowpitehrtdieastaanshdoawssnemasbcliyrcelfefisc(ieBnMcyV. )(,as)qSucaarleinsg(HofPtVma)xawnidth

hexagons (HBV). (b) data are for HBV and

The reaction order for nucleation, HPV. Note that discontinuities in

tnhce,

is obtained from the scaling experimental kinetic traces

fboerhHavPiVoraosfsermmaxb.lTyhe

abreetwreesepnoenlsoinbgleatfoiorninaancdcunruacclieeastiionndientetrhme ivnirinalgsrymsatxemfosr

data at higher initial studied in this work.

concentrations. (c) Balance Green solid line corresponds

to by

ε = 4.9 × 10−5, green dashed lines circles, HPV by squares, HBV by

hcoexrraegsopnosnadntdoHεcB(VEqa.s(s1e4m)b) lfyordtahtae

various viruses. BMV data denoted obtained at 37 °C and pH 7.5 from50

(see Supplementary Information) by triangles. Blue data are from actin polymerization measurements in

magnesium (stars) or in calcium (diamonds) from51. Viral images reproduced from23 with permission.

the underlying kinetic equations illuminates the dynamic design criteria that characterize the efficiency of such processes. We showed that efficient assembly only occurs in a narrow range of parameter space. By applying this kinetic analysis to experimental data of icosahedral viral capsid assembly we demonstrated that these structures occupy this narrow region of parameter space corresponding to efficient assembly.
References
	 1.	 Whitesides, G. M. & Grzybowski, B. Self-assembly at all scales. Science 295, 2418–2421 (2002). 	 2.	 Edeling, M. A., Smith, C. & Owen, D. Life of a clathrin coat: insights from clathrin and AP structures. Nat. Revs. Mol. Cell Biol. 7,
32–44 (2006). 	 3.	 Fletcher, J. M. et al. Self-assembling cages from coiled-coil peptide modules. Science 340, 595–599 (2013). 	 4.	 Fath, S., Mancias, J. D., Bi, X. & Goldberg, J. Structure and organization of coat proteins in the COPII cage. Cell 129, 1325–1336
(2007). 	 5.	 Boato, F. et al. Synthetic virus-like particles from self-assembling coiled-coil lipopeptides and their use in antigen display to the
immune system. Angew. Chem. Int. Ed. 46, 9015–9018 (2007). 	 6.	 Wörsdörfer, B., Woycechowsky, K. J. & Hilvert, D. Science 331, 589–592 (2011). 	 7.	 King, N. P. et al. Computational design of self-assembling protein nanomaterials with atomic level accuracy. Science 336, 1171–1174
(2012). 	 8.	 Lai, Y.-T., Cascio, D. & Yeates, T. O. Structure of a 16-nm cage designed by using protein oligomers. Science 336, 1129 (2012). 	 9.	 Caspar, D. L. & Klug, A. Physical principles in the construction of regular viruses. Cold Spring Harb Symp Quant Biol 27, 1–24
(1962).

Scientific Reports | 7: 12295 | DOI:10.1038/s41598-017-12528-8

6

www.nature.com/scientificreports/

	10.	 Bancroft, J. B., Hills, G. J. & Markham, R. A study of the self-assembly process in a small spherical virus. Formation of organized structures from protein subunits in vitro. Virology 31, 354–379 (1967).
	11.	 Crick, F. H. C. & Watson, J. D. Structure of small viruses. Nature 177, 473–475 (1956). 	12.	 Fasching, L. et al. TRIM28 represses transcription of endogenous retroviruses in neural progenitor cells. Cell Rep 10, 20–28 (2015). 	13.	 Knowles, T. P. J., Vendruscolo, M. & Dobson, C. M. The amyloid state and its association with protein misfolding diseases. Nat. Rev.
Mol. Cell Biol. 15, 384–96 (2014). 	14.	 Oosawa, F. & Asakura, S. Thermodynamics of the polymerisation of proteins (Academic Press, 1975). 	15.	 Knowles, T. P. J. et al. An analytical solution to the kinetics of breakable filament assembly. Science 326, 1533–1537 (2009). 	16.	 Cohen, S. I. A. et al. Proliferation of amyloid-β42 aggregates occurs through a secondary nucleation mechanism. Proc. Natl. Acad.
Sci. USA 110, 9758–9763 (2013). 	17.	 Meisl, G. et al. Molecular mechanisms of protein aggregation from global fitting of kinetic models. Nat. Protoc. 11, 252–272 (2016). 	18.	 Thomas C. T. Michaels et al. Hamiltonian Dynamics of Protein Filament Formation. Phys Rev Lett 116, 038101 (2016). 	19.	 Zlotnick, A., Johnson, J. M., Wingfield, P. W., Stahl, S. J. & Endres, D. A theoretical model successfully identifies features of Hepatitis
B virus capsid assembly. Biochemistry 38, 14644–14652 (1999). 	20.	 Endres, D. & Zlotnick, A. Model-based analysis of assembly kinetics for virus capsids or other spherical polymers. Biophys J 83,
1217–1230 (2002). 	21.	 Casini, G. L., Graham, D., Heine, D., Garcea, R. L. & Wu, D. T. In vitro papillomavirus capsid assembly analyzed by light scattering.
Virology 325, 320–327 (2004). 	22.	 Chen, C., Kao, C. C. & Dragnea, B. Self-assembly of Brome Mosaic virus capsids: Insights from shorter time-scale experiments. J
Phys Chem A 112, 9405–9412 (2008). 	23.	 Fauquet, C. M., Mayo, M. A., Maniloff, J., Desselberger, U. & Ball, L. A. Eds Virus Taxonomy: VIIIth Report of the International
Committee on Taxonomy of Viruses, (Elsevier/Academic Press, 2005). 	24.	 Prevelige, P. E. Jr., Thomas, D. & King, J. Nucleation and growth phases in the polymerization of coat and scaffolding subunits into
icosahedral procapsid shells. Biophys J 64, 824–835 (1993). 	25.	 Michaels, T. C. T. et al. Phys Rev Lett 116, 258103 (2016). 	26.	 Zlotnick, A., Aldrich, R., Johnson, J. M., Ceres, P. & Young, M. J. Virology 277, 450 (2000). 	27.	 Johnson, J. M. et al. Nano Lett. 5, 765–770 (2005). 	28.	 Chen, L.-Y., Goldenfeld, N. & Oono, Y. Renormalization group theory for global asymptotic analysis. Phys Rev Lett 73, 1311–1315
(1994). 	29.	 Barabasi, A. L. & Albert, R. Emergence of scaling in random networks. Science 286, 509–512 (1999). 	30.	 Skotheim, J. M. & Mahadevan, L. Physical limits and design principles for plant and fungal movements. Science 308, 1308–1310
(2005). 	31.	 Zhang, T. & Schwartz, R. Simulation study of the contribution of oligomer/oligomer binding to capsid assembly kinetics. Biophys J
90, 57–64 (2006). 	32.	 Keef, T., Micheletti, C. & Twarock, R. Master equation approach to the assembly of viral capsids. J Theor Biol 242, 713–721 (2006). 	33.	 Hemberg, M., Yaliraki, S. N. & Barahona, M. Stochastic kinetics of viral capsid assembly based on detailed protein structures.
Biophys J 90, 3029–3042 (2006). 	34.	 Smith, G. R., Xie, L., Lee, B. & Schwartz, R. Applying molecular crowding models to simulations of virus capsid assembly in vitro.
Biophys J 106, 310–320 (2014). 	35.	 Dykeman, E. C., Stockley, P. G. & Twarock, R. Building a viral capsid in the presence of genomic RNA. Phys Rev E 87(022717), 1–12
(2013). 	36.	 Hagan, M. F. Modeling viral capsid assembly. Adv Chem Phys 155, 1–67 (2014). 	37.	 Zandi, R., van der Schoot, P., Reguera, D., Kegel, W. & Reiss, H. Classical nucleation theory of virus capsids. Biophys J 90, 1939–1948
(2006). 	38.	 van der Schoot, P. & Zandi, R. Kinetic theory of virus capsid assembly. Phys Biol 4, 296–304 (2007). 	39.	 Morozov, A. Y., Bruinsma, R. F. & Rudnick, J. Assembly of viruses and the pseudo-law of mass action. J Chem Phys 131(155101),
1–17 (2009). 	40.	 Hagan, M. F. & Chandler, D. Dynamic pathways for viral capsid assembly. Biophys J 91, 42–54 (2006). 	41.	 Matthews, R. & Likos, C. N. Dynamics of self-assembly of model viral capsids in the presence of a fluctuating membrane. J Phys
Chem B 117, 8283–8292 (2013). 	42.	 Johnston, I. G., Louis, A. A. & Doye, J. P. K. Modelling the self-assembly of virus capsids. J Phys: Condens Matter 22(104101), 1–9
(2010). 	43.	 Hicks, S. D. & Henley, C. L. Irreversible growth model for virus capsid assembly. Phys Rev E 74(031912), 1–17 (2006). 	44.	 Nguyen, H. D., Reddy, V. S. & Brooks, C. L. Deciphering the kinetic mechanism of spontaneous self-assembly of icosahedral capsids.
Nano Lett 7, 338–344 (2007). 	45.	 Nguyen, H. D., Reddy, V. S. & Brooks, C. L. Invariant polymorphism in virus capsid assembly. J Am Chem Soc 131, 2606–2614
(2009). 	46.	 Schwartz, R., Shor, P. W., Prevelige, P. E. & Berger, B. Local rules simulation of the kinetics of virus capsid self-assembly. Biophys J 75,
2626–2636 (1998). 	47.	 Wilber, A. W. et al. Reversible self-assembly of patchy particles into monodisperse icosahedral clusters. J Chem Phys 127(085106),
1–11 (2007). 	48.	 Rapaport, D. C. Molecular dynamics simulation of reversibly self-assembling shells in solution using trapezoidal particles. Phys Rev
E 86(051917), 1–7 (2012). 	49.	 Perlmutter, J. D. & Hagan, M. F. Mechanisms of virus assembly. Annu Rev Phys Chem 66, 217–239 (2015). 	50.	 Selzer, L., Katen, S. P. & Zlotnick, A. The Hepatitis B virus core protein intradimer interface modulates capsid assembly and stability.
Biochemistry 53, 5496–5504 (2014). 	51.	 Cooper, J. A., Buhle, E. L. Jr., Walker, S. B., Tsong, T. Y. & Pollard, T. D. Kinetic evidence for a monomer activation step in actin
polymerization. Biochemistry 22, 2193–2202 (1983). 	52.	 Pierson, E. E. et al. Detection of late intermediates in virus capsid assembly by charge detection mass spectrometry. J. Am. Chem. Soc.
136, 3536–41 (2014). 	53.	 Packianathan, C., Katen, S. P., Dann, C. E. 3rd & Zlotnick, A. Conformational changes in the hepatitis B virus core protein are
consistent with a role for allostery in virus assembly. J. Virol. 84, 1607–15 (2010). 	54.	 Lazaro, G. R. & Hagan, M. F. Allosteric Control of Icosahedral Capsid Assembly. J. Phys. Chem. B 120, 6306–18 (2016). 	55.	 Hagan, M. F. & Elrad, O. M. Understanding the concentration dependence of viral capsid assembly kinetics-the origin of the lag time
and identifying the critical nucleus size. Biophys J 98, 1065–1074 (2010).
Acknowledgements
We acknowledge support from the Swiss National Science Foundation (TCTM), Peterhouse, Cambridge (TCTM), the Cambridge Commonwealth, European and International Trust (MMJB), the NIH-Oxford/ Cambridge Scholars Program (MMJB), Award Number R01GM108021 from the National Institute of General

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Medical Sciences (MFH), ERC (TCTM, TPJK), BBSRC (TPJK) and the Newman foundation (TPJK). This work was in part supported by the Intramural Research Program of the National Institute of Digestive and Diabetes and Kidney Diseases at the National Institutes of Health (MMJB). We thank C. Chen for providing the data in ref.22. We thank A. Zlotnick and L. Seltzer for kindly providing the data in refs19,50 and for fruitful discussions. We thank R.B. Best, R.L. Jack, G. Meisl and C.M. Dobson for useful comments and discussions.
Author Contributions
All authors conceived and designed the project; T.C.T.M. developed the theory; T.C.T.M. and M.M.J.B. analysed the experimental data; all authors wrote the paper.
Additional Information
Supplementary information accompanies this paper at https://doi.org/10.1038/s41598-017-12528-8. Competing Interests: The authors declare that they have no competing interests. Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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