Mon e Ca lo Simula ions o Disc e e Rouse
Dynamics on a 2D La ice
Eme gence o Global Beha io in a Polyme
Chain om Local Cons ain s
A pan Dey
M1 Physics (IDIL)
Uni e si é de Mon pellie
Acknowledgmen s
I would like o exp ess my since e hanks o D . Jean-Cha les Wal e ,
P o . And ea Pa meggiani and Linda Delimi o hei guidance and
suppo in he p epa a ion and e inemen o his epo . This epo was
p epa ed du ing my bibliog aphic s udies in ad ance o my in e nship a he
Labo a oi e Cha les Coulomb (L2C), Uni e si é de Mon pellie .
Table o Con en s
Abs ac
1 The Rouse Model
2 Mon e Ca lo Implemen a ion o
Disc e e Rouse Dynamics
3 Polyme Dynamics unde
He e ogeneous Geome ic Rules
Discussion and Conclusion
Re e ences
Appendix: End-Monome
In e ac ions as a Minimal Model
o Polyme Collapse
Abs ac
A polyme , in simple wo ds, is a connec ed chain o indi idual uni s called monome s. Unlike
gas pa icles, polyme s a e connec ed chains, which makes he modeling o polyme
in e ac ions mo e di icul – each monome is connec ed o he monome s immedia ely be o e
and a e i along he chain, which pu s cons ain s on he ways he monome s (and hence he
polyme ) can mo e, e en wi hou conside ing ex e nal e ec s om he su ounding. The Rouse
model p o ides a minimal heo e ical amewo k o s udying polyme dynamics. I desc ibes
monome mo ion as ha monically coupled B ownian oscilla o s ( es o ing o ce plus andom
he mal noise!). In his epo , we explo e a disc e e, la ice-based analogue o he Rouse model
by en o cing a single local cons ain – p ese a ion o he nea es -neighbo dis ances along
he polyme chain – on a wo-dimensional la ice, along wi h sel -a oidance. Using Mon e
Ca lo simula ions, we s udy he e olu ion o he chain unde his cons ain . We quan i y his
beha io by acking he squa ed end- o-end dis ance 𝑅𝑒
2, he squa ed adius o gy a ion 𝑅𝑔
2 and
he mean squa ed displacemen ⟨|𝑟𝑖(𝑡)−𝑟𝑖(0)|2〉, he la e exhibi ing a cha ac e is ic
subdi usi e (~ 𝑡1/2) o di usi e (~ 𝑡) c osso e a a imescale o he o de o 𝑁2 i we d op
sel -a oidance (𝑁 is he numbe o monome s). This obse a ion is consis en wi h Rouse
scaling. Despi e i s simplici y and lack o explici ene gy e ms, he disc e e dis ance-
p ese ing ule encapsula es essen ial aspec s o polyme dynamics, illus a ing how complex
global beha io can eme ge om minimal mic oscopic cons ain s.
To u he explo e he beha io o polyme s on a 2D la ice beyond Rouse dynamics, we
in oduce a se o al e na ing- ule oy models (al e na ing copolyme s) and a block copolyme
oy model o illus a e how simple, spa ially he e ogeneous geome ic cons ain s can d i e he
sys em away om de ailed balance. By modi ying only he kinds o mo es di e en monome s
a e pe mi ed o a emp – wi hou in oducing o ces, po en ials o ene ge ic biases – we c ea e
a amily o simple, nonequilib ium models ha gene a e di e se, global beha io . In some
models, de ia ions om s anda d Rouse dynamics a e e y weak, because geome ic
cons ain s supp ess he he e ogeneous upda e ules; o he s p oduce s ong nonequilib ium
expansion o polyme s, d i en by luc ua ions in he bond leng hs. Toge he , hese esul s show
ha nonuni o m geome ic ules can encode a wide spec um o dynamical esponses while
e aining concep ual simplici y. O e all, his epo demons a es ha complex global
phenomena in polyme dynamics can eme ge om minimal local cons ain s, and ha la ice
models p o ide an ins uc i e pla o m o dissec ing how equilib ium and nonequilib ium
ea u es a ise om pu ely geome ic p inciples.
Compu a ional no e: The Mon e Ca lo simula ions pe o med in ol e a la ge numbe o
upda e a emp s (up o he o de o 107) pe un and la ge ensembles (up o 100 independen
ealiza ions pe chain leng h). Execu ing his se ially, ei he locally o on b owse -based
pla o ms such as Google Colab, becomes ex emely slow and imp ac ical due o single-co e
execu ion and p ocess o e head. Thus, he simula ions we e un locally using Py hon ins alled
ia he Mini o ge Conda dis ibu ion, and pa allelized using Py hon’s mul ip ocessing module,
wi h each un execu ed on a sepa a e CPU co e. This app oach p ese es iden ical Mon e Ca lo
dynamics while g ea ly educing o al un ime.
The Rouse Model
The Rouse model p o ides one o he simples desc ip ions o polyme dynamics. I ea s a
polyme chain as a sequence o 𝑁 monome s connec ed by ha monic sp ings. Each monome
is imme sed in a iscous medium and expe iences wo opposing in luences: ic ion om he
su ounding luid and andom he mal kicks due o molecula collisions. The combina ion o
hese o ces gi es ise o s ochas ic mo ion o he en i e chain (B ownian oscilla o s!).
In i s con inuum o m, he ime e olu ion o he posi ion 𝑟𝑖(𝑡) o he 𝑖- h monome is gi en by
he equa ion:
𝜁𝑑𝑟𝑖
𝑑𝑡 =𝑘(𝑟𝑖+1+𝑟𝑖−1−2𝑟𝑖)+𝜉𝑖(𝑡)
whe e 𝜁 is he ic ion coe icien , 𝑘 is he sp ing cons an , and 𝜉𝑖(𝑡) ep esen s he andom
he mal o ces wi h ze o mean. The e m in ol ing he neighbo ing monome s (𝑟𝑖−1 and 𝑟𝑖+1)
exp esses a ha monic es o ing o ce – each monome beha es like a coupled oscilla o ,
connec ed o i s neighbo s, while s ill subjec o andom he mal mo ion.
The Rouse model o iginally assumes he polyme o be an ideal Gaussian chain. In simple
wo ds, his means we allow mo e han one monome o “o e lap” – hey do no epel each
o he and do no collide (no olume exclusion). Thus, e en wo dis an pa s o he chain a e
allowed – ma hema ically speaking – o pass a bi a ily close o o e lap in space. The Rouse
equa ions desc ibe a long phan om chain ha can eely c umple and old wi hou ge ing in i s
own way – some hing ha is, o cou se, no ue o eal polyme s bu s ill accu a ely cap u es
many la ge-scale dynamical p ope ies.
Unde his assump ion, each s ep o he polyme (basically he ec o om monome 𝑖 o
monome 𝑖+1) – can ea ed as an independen andom s ep. The e is no p e e ed di ec ion
and no co ela ion be ween di e en s eps. Thus, he ideal polyme jus ollows he s a is ics o
a andom walk. I we ake 𝑁 such s eps, each o leng h 𝑎, he end- o-end ec o o his 𝑁-
segmen beha es like a sum o independen andom ec o s. We de ine he s eps using he s ep
ec o Δ𝑟𝑖, 𝑖=1,2…𝑁, each ec o o ixed size |Δ𝑟𝑖|=𝑎. The a e age o each s ep is ze o
and s eps a e unco ela ed, hence we es ima e he squa e o he size o he segmen . As a
eminde , we do no assume sel -a oidance, bu we assume he s eps a e independen o
di ec ion and iden ical. Then he end- o-end ec o would be (please mind he no a ion!):
Δ𝑟=∑Δ𝑟𝑖
𝑁
𝑖=1
Since we ha e assumed iso opy and independence o s eps, we expec he a e age
displacemen o be ze o: ⟨Δ𝑟𝑖〉=0. We also ha e:
⟨Δ𝑟𝑖.Δ𝑟𝑗〉=𝑎2𝛿𝑖𝑗
𝛿𝑖𝑗 is he K onecke del a, which means he mean o he do p oduc o 𝑟𝑖 and 𝑟𝑗 is always ze o,
excep when 𝑖=𝑗 (when we a e alking abou he same s ep, o he same monome on he
la ice), in which case hei do p oduc yields 𝑎2, as expec ed. This does no mean ha when
𝑖≠𝑗, he 𝑖- h and 𝑗- h s eps a e o hogonal – i simply means he e is no co ela ion be ween
hem o e a la ge numbe o s eps (since we a e conside ing he mean o he do p oduc ).
Gi en ha we ha e assumed iso opy and independence, his makes sense. Now, keeping all
ha in mind, le us calcula e he mean squa ed end- o-end dis ance o he chain:
⟨Δ𝑟2〉=⟨Σ𝑖Δ𝑟𝑖.Σ𝑗Δ𝑟𝑗〉=Σ𝑖,𝑗⟨Δ𝑟𝑖.Δ𝑟𝑗〉=Σ𝑖𝑎2=𝑎2𝑁
This is cha ac e is ic o Gaussian polyme s: he a iance o he end- o-end ec o ⟨Δ𝑟2⟩ g ows
linea ly wi h he numbe o s eps 𝑁. This means an ideal polyme does no s e ch ou s i ly;
i meande s. I you zoom in on any chunk o monome s, ha chunk pe o ms a minia u e
andom walk o i s own. Doubling he numbe o s eps 𝑁 makes he mean-squa ed size wice
as la ge, and so on; and so he ypical linea size g ows only as √𝑁. This squa e- oo scaling
essen ially se s he na u al leng h scale o any in e nal mo ion o he chain. I some po ion o
he polyme has had ime o in e nally elax (lose co ela ions wi h i s neighbo s), he spa ial
luc ua ions i explo es a e o he o de Δ𝑟(𝑁)∼𝑁1/2, no 𝑁 i sel . The Gaussian iewpoin
makes he Rouse model sol able and unde pins all i s scaling laws. I is impo an o keep in
mind ha in his pic u e, he chain’s in e nal “wiggles” ha e he same s a is ical s uc u e as a
andom walk, and he dynamics simply go e n how as di e en pa s o ha walk can
ea ange.
In his wo k, we educe he con inuous Rouse model o a disc e e, la ice-based e sion ha
s ill p ese es i s essen ial physical idea. Ins ead o explici ha monic o ces o s ochas ic
di e en ial equa ions, we en o ce a single geome ic ule – each monome can mo e only in
ways ha p ese e he dis ance o i s nea es neighbo s along he chain. This disc e e cons ain
e ec i ely eplaces he ha monic po en ial wi h a dis ance-p ese ing ule. Fu he , we also
d op he assump ion ha monome s can o e lap, and inco po a e olume exclusion in ou
model. Using Mon e Ca lo simula ions on a 2D la ice, we s udy how such pu ely local mo es
gi e ise o eme gen la ge-scale beha io s consis en wi h Rouse dynamics, including
elaxa ion om s e ched and andom ini ial con igu a ions, luc ua ions in polyme size
( h ough bo h he end- o-end dis ance and he adius o gy a ion) and he subdi usi e- o-
di usi e c osso e in monome mo ion.
We can mo e he polyme chain (we a e alking abou a Gaussian polyme now!) in many
di e en ways – s e ch i , wiggle i in he middle, wiggle only one end – and he small, local
dis u bances ade away quickly, whe eas he la ge , slowe dis u bances sp ead h oughou he
polyme and ake ime o die ou . Thus, he e a e many elaxa ion imes o he di e en kinds
o mo ion, and he slowes o hese elaxa ion imes g ow as 𝑁2, whe e 𝑁 is he leng h o he
polyme ( he numbe o monome s): 𝜏∝𝑁2
This is simply he o de o ime i akes o he polyme o comple ely “ o ge ” i s ini ial
con igu a ion. This is isualized by plo ing ⟨|𝑟𝑖(𝑡)−𝑟𝑖(0)|2〉 – he mean squa ed displacemen
o an a bi a y monome 𝑖 (no monome is special in he chain, so his monome is a
ep esen a i e o he a e age monome ) – o e ime. E en hough we a e acking a single
monome , his gi es a measu e o how much he polyme has mo ed om i s ini ial
con igu a ion ( ime 𝑡=0) a some la e ime 𝑡=𝑡. A sho imes, he monome s ha e a s ong
memo y o he ini ial con igu a ions, and he mo ion o a single monome is subdi usi e – he
polyme canno di use eely since he mo ion o he monome s a e hea ily cons ained by he
connec ions o he neighbo ing monome s. The mean squa e displacemen , in his egime,
scales as:
⟨Δ𝑟2(𝑡)⟩∼𝑡1/2
O e a su icien ly long pe iod o ime, he polyme as a whole di uses eely:
⟨Δ𝑟2(𝑡)⟩∼𝑡
The c osso e om he subdi usi e o di usi e egime akes place a a imescale o abou 𝑁2.
Fi s , we unde s and why ⟨Δ𝑟2(𝑡)⟩ scales as 𝑡1/2 o small imes. We ocus on an a bi a y
monome in he bulk o he polyme (no a he edges). When his monome is pe u bed by
andom he mal luc ua ions, i sligh ly “d ags” i s neighbo s (sp ings pull!). Those neighbo s,
in u n, d ag hei neighbo s and so on. O e ime, he ini ial monome e ec i ely ge s coupled
o a g owing chunk o he chain a ound i . Le he size o his chain, which is a unc ion o ime,
be 𝑁(𝑡).
In he Rouse model, he dynamics o each monome a e o e damped because he su ounding
sol en exe s a s ong iscous d ag ha domina es o e ine ial e ec s ( he e m 𝜁 in he
o iginal equa ion). A he mic oscopic scale, a monome is iny and does no accumula e
momen um – any signi ican displacemen is immedia ely opposed by a ic ional o ce
p opo ional o i s eloci y, leading o slow elaxa ional mo ion a he han linea wa e-like
p opaga ion. As a consequence, a local change in he posi ion o one monome is ansmi ed
o i s neighbo s only g adually, in a way ha is ma hema ically analogous o a di usi e p ocess
along he monome s (jus like he di usion equa ion, he Rouse di e en ial equa ion con ains
a i s o de ime de i a i e and a second o de space de i a i e – he sp ing o ces e m! –
along he monome index). The in luence o a dis u bance sp eads ou wa d wi h he
cha ac e is ic di usion law 𝑁(𝑡)∝√𝑡. Thus, a e ime 𝑡, only a segmen o oughly 𝑁(𝑡)
monome s a ound he agged bead has esponded o he pe u ba ion.
Now, as we ha e seen using andom walk s a is ics: Δ𝑟2 ~ 𝑎2𝑁. A e ime 𝑡, only he 𝑁(𝑡)-
monome chunk has had ime o espond o he luc ua ions o he ini ial monome . So he
mean-squa e displacemen o he ini ial monome a e ime 𝑡 is o he same o de as he
equilib ium a iance o ha chunk:
⟨Δ𝑟2⟩=𝑎2𝑁 ~ 𝑎2√𝑡∝𝑡1/2
So he ypical displacemen scales as:
Δ𝑟(𝑡) ~ 𝑡1/4
This is he dis ance a monome mo es due o in e nal chain mo ions be o e he chain s a s
d i ing collec i ely.
We ha e al eady seen ha , o a andom walk polyme , ⟨Δ𝑟2〉=𝑎2𝑁. To es ima e how he size
o his andom walk polyme scales, we look a he squa e- oo o ⟨Δ𝑟2〉:
√⟨Δ𝑟2〉 ~ √𝑁 ~ 𝑁1/2
Thus, he size o an ideal polyme scales as Δ𝑟 ~ 𝑁1/2 unde andom walk s a is ics. This
means he monome loses “memo y” o i s ini ial posi ion only when i has di used oughly
he ull chain size:
Δ𝑟(𝑡) ~ 𝑁1/2
Subs i u e his esul in Δ𝑟(𝑡) ~ 𝑡1/4, we ge :
Δ𝑟(𝑡) ~ 𝑡1/4 ~ 𝑁1/2
I is impo an o no e ha Δ𝑟(𝑡) ~ 𝑡1/4 holds in he subdi usi e egime, and since 𝑁1/2 is a
measu e o he ull size o he chain, Δ𝑟(𝑡) ~ 𝑁1/2 indica es ha he monome has al eady
di used h ough he en i e chain, and hence ully los he “memo y” o i s ini ial posi ion.
Hence, he o de o ime we ge by compa ing 𝑡1/4 and 𝑁1/2 would gi e us a measu e o he
imescale o he c osso e o he polyme om subdi usi e o di usi e egime:
𝑡1/4 ~ 𝑁1/2 → 𝑡 ≡𝜏 ~ 𝑁2
A imescales 𝜏 ~ 𝑁2, he monome ’s local subdi usi e mo ion has ca ied i a dis ance
compa able o he o e all chain size, and he polyme di uses eely since all “memo y” o he
ini ial con igu a ion is los . Essen ially wi h he passage o ime, he mo e he polyme
“ elaxes” ( he a he he polyme mo es om i s ini ial con igu a ion), he mo e eely i can
di use – his is one o he de ining ea u es o he Rouse model.
A i s glance, i may seem coun e in ui i e ha sho e polyme s (small 𝑁) di use mo e
apidly (since 𝜏 ~ 𝑁2). Monome s in a sho e chain a e mo e s ongly co ela ed, so we would
expec hei mo ions igh ly cons ained, and he polyme should mo e mo e slowly and ake
mo e ime o di use. Howe e in he Rouse pic u e, hese co ela ions ac ually make he en i e
chain mo e oge he mo e cohe en ly and di use as e . The o al ic ion expe ienced by he
polyme scales wi h he numbe o monome s 𝑁, so he di usion coe icien o he cen e o
mass o he polyme is o he o de :
𝐷𝑐𝑚 ∼1
𝑁
A sho e polyme , ha ing ewe monome s, expe iences less o al d ag and hus di uses mo e
easily as a whole. While longe chains exhibi slowe , less coo dina ed mo ion due o in e nal
luc ua ions, sho e chains beha e almos like a single igid body, leading o as e o e all
di usion despi e hei s onge in e nal co ela ions.
We can in ui i ely y o unde s and why 𝐷𝑐𝑚 scales as 1/𝑁. Fo a polyme consis ing o 𝑁
monome s, he cen e o mass 𝑟𝑐𝑚 is gi en by:
𝑟𝑐𝑚 =1
𝑁∑𝑟𝑖
𝑁
𝑖=1
Each monome eels some ic ion ( ic ion coe icien 𝜁) om he su ounding medium and
andom he mal noise 𝜉𝑖(𝑡). The Rouse equa ion o one bead is:
𝜁𝑑𝑟𝑖
𝑑𝑡 =(sp ing o ces) +𝜉𝑖(𝑡)
Now, we conside all 𝑁 monome s and sum o e all he equa ions. All he in e nal sp ing o ces
cancel – e e y sp ing ac s equal and opposi e on i s neighbo s. This lea es us wi h:
𝜁∑𝑑𝑟𝑖
𝑑𝑡
𝑖=∑𝜉𝑖(𝑡)
𝑖
Using 𝑟𝑐𝑚 =1
𝑁∑𝑟𝑖
𝑁
𝑖=1 , we may ew i e he le hand side o he abo e exp ession:
𝑁𝜁𝑑𝑟𝑐𝑚
𝑑𝑡 =∑𝜉𝑖(𝑡)
𝑖
Now, each andom he mal o ce 𝜉𝑖(𝑡) is independen , wi h a iance 𝜎2 (say).
When we sum 𝑁 o hem, he a iances add up and he a iance o he o al noise is he
indi idual a iance mul iplied by 𝑁:
⟨(Σi𝜉𝑖−⟨Σi𝜉𝑖〉)2⟩∝𝑁𝜎2
Since o he mal noise ⟨Σi𝜉𝑖〉=0, we ge :
⟨(Σi𝜉𝑖)2⟩∝𝑁𝜎2
This means:
Σi𝜉𝑖 ~ √𝑁𝜎
Using his esul in 𝑁𝜁𝑑𝑟𝑐𝑚
𝑑𝑡 =∑ 𝜉𝑖(𝑡)
𝑖, we ge :
𝑁𝜁𝑑𝑟𝑐𝑚
𝑑𝑡 ~ √𝑁𝜎
F om his, we see ha he ypical eloci y o he cen e o mass is:
𝑣𝑐𝑚 =𝑑𝑟𝑐𝑚
𝑑𝑡 ~√𝑁𝜎
𝑁𝜁
In he abo e exp ession, √𝑁𝜎 is a measu e o he ypical ampli ude o noise ha he polyme
is subjec ed o – he noise pushes he pa icle away om i s ini ial posi ion and con ibu es o
i s di usion. The denomina o 𝑁𝜁 ep esen s he ic ion, which is ob iously in e sely ela ed
o he eloci y.
We nex de ine a noise co ela ion ime 𝜏𝑐, which is simply he maximum ime o which he
noise emains co ela ed ( o imes g ea e han 𝜏𝑐, he sys em can be ea ed o be independen
o ea lie andom kicks). In o he wo ds, he polyme akes an unco ela ed s ep a e ime
in e als o 𝜏𝑐. O e he co ela ion ime 𝜏𝑐, he displacemen o he cen e o mass would scale
as: Δ𝑟𝑐𝑚 ∼𝑣𝑐𝑚𝜏𝑐
And hus, he mean squa ed displacemen pe kick scales as:
⟨(Δ𝑟𝑐𝑚)2〉 ~ 𝑣𝑐𝑚
2𝜏𝑐
2
➢ I i is an end monome , allow i o “o bi ” i s neighbo by one la ice s ep, while
espec ing sel -a oidance (end lips)
➢ I i is no an end monome , c ea e a lis o he si es ha keep bo h neighbo bonds
a leng h 1
➢ Re u n hose candida e posi ions o he Mon e Ca lo s ep o choose om
(equip obably)
5. Once he mo e has been decided, upda e he con igu a ion and epea (and gene a e isuals
o he polyme chain on he 2D la ice a egula in e als, as speci ied by he plo in e al).
Fi s , we s a wi h a s aigh chain ( ully s e ched ini ial con igu a ion) – as a es case. We
ha e chosen la ice size 𝐿=70 (so we ha e a 70X70 la ice, and 4900 la ice si es in o al),
𝑁=50 ( he numbe o monome s, o leng h o he polyme ), numbe o s eps = 5000, plo
in e al = 100. We look a some snapsho s a di e en Mon e Ca lo s eps o isualize he
beha io o he polyme .
Figu e 2: Mon e Ca lo simula ion o a polyme chain o leng h 50 unde disc e e Rouse dynamics on
70X70 2D la ice o s aigh ini ial con igu a ion
S a ing om a ully s e ched (s aigh ) ini ial con igu a ion, we see ha he polyme
unde goes isible elaxa ion o e successi e Mon e Ca lo s eps, g adually adop ing mo e
compac and i egula con o ma ions. E en hough we do no ha e any explici a ac i e
po en ial he e, he chain appea s o “sh ink” sligh ly as i explo es con igu a ions ha sa is y
he nea es -neighbo dis ance cons ain while a oiding sel -in e sec ions. This e lec s he
en opic endency o he polyme o sample a wide ange o con o ma ions a he han emain
ex ended. I should also be ecalled ha he simula ion is inhe en ly s ochas ic – a each s ep,
each monome mo e is selec ed andomly om he se o allowed op ions – and hence we
would ge sligh ly di e en con o ma ions i we e un he same simula ion wi hou changing
any o he pa ame e alues. Howe e , we would s ill obse e a simila o e all elaxa ion
beha io .
We now sligh ly change he pa ame e s: 𝐿=50, 𝑁=30, and s a wi h a ully s e ched
con igu a ion like be o e:
Figu e 3: Mon e Ca lo simula ion o a polyme chain o leng h 30 unde disc e e Rouse dynamics on
50X50 2D la ice o s aigh ini ial con igu a ion
When a sho e polyme chain (𝑁=30) is simula ed unde he same Mon e Ca lo condi ions,
we see ha he o e all elaxa ion occu s mo e apidly and he chain displays mo e p onounced
bending wi hin he same numbe o s eps. This is no su p ising, because sho e chains ha e
ewe in e nal deg ees o eedom and hence, s onge co ela ions wi h he neighbo ing
monome s – any local mo e a ec s a signi ican po ion o he en i e polyme , e ec i ely
bending i quickly and app eciably e en i he ini ial con igu a ion was ully s aigh . The
la ice size (𝐿=50) plays no signi ican ole he e, since i emains much la ge han he
polyme i sel (we ha e chosen 𝐿=50 he e ins ead o 𝐿=70 o isual cla i y).
We now isualize he same polyme as abo e (𝑁=30, 𝐿=50), bu a e emo ing he
cons ain o olume exclusion. We clea ly see ha he Gaussian polyme o e laps wi h and
in e sec s i sel , and o he same numbe o Mon e Ca lo s eps he Gaussian polyme collapses
o a much g ea e ex en han he sel -a oiding polyme .
Figu e 4: Mon e Ca lo simula ion o a Gaussian polyme chain o leng h 30 unde disc e e Rouse
dynamics on 50X50 2D la ice o s aigh ini ial con igu a ion
Le us now un he simula ion s a ing om a andom ini ial con igu a ion. This ime, we
choose an in e media e polyme leng h 𝑁=40, and since we a e s a ing om a andom (ben )
con igu a ion, we choose a sligh ly smalle la ice o be e isual cla i y, 𝐿=30. We keep he
numbe o s eps = 5000 and plo in e al = 100. When gene a ing a andom ini ial
con igu a ion, we mus ensu e o a oid si ua ions in which he gene a ing p ocess becomes
apped in a dead end. In he p ocedu e used he e, he chain is ini ia ed nea he cen e o he
la ice and ex ended s ep-by-s ep in bo h di ec ions by choosing an a ailable nea es -neighbo
si e (acco ding o he ule o p ese ing nea es -neighbo dis ances). Al hough his me hod
usually succeeds o mode a e chain leng hs, i is s ill possible o he g ow h o e mina e
p ema u ely ( o example, we may speci y 𝑁=100, bu he p ocess e mina es a 𝑁=93).
This is especially impo an in wo dimensions – due o he limi ed a ailabili y o ee
neighbo ing si es in 2D, sel -a oidance s ongly es ic s he space o accessible con igu a ions.
G owing he chain ou wa d in bo h di ec ions educes he likelihood o ge ing apped, bu i
does no elimina e he isk en i ely, and i is always a good idea o check ha he algo i hm
success ully gene a es a ull-leng h con igu a ion.
Figu e 5: Mon e Ca lo simula ion o a polyme chain o leng h 40 unde disc e e Rouse dynamics on
30X30 2D la ice o andom ini ial con igu a ion
We see ha o a andom ini ial con igu a ion, he polyme unde goes a mo e equilib a ed
e olu ion, showing no s ong endency o ei he swell o collapse. The polyme keeps explo ing
a wide ange o con o ma ions, luc ua ing a ound an a e age size de e mined by he balance
be ween connec i i y cons ain s and sel -a oidance. The absence o any ene ge ic bias means
he dynamics a e pu ely en opic – he chain ea anges locally while p ese ing bond leng hs
and a oiding o e laps. We now epea he abo e simula ion o a Gaussian polyme wi h 𝑁=
40, 𝐿=30 as abo e. This ime, we only show h ee snapsho s ins ead o six.
Figu e 6: Mon e Ca lo simula ion o a Gaussian polyme chain (N=40, L=30) unde disc e e Rouse
dynamics o andom ini ial con igu a ion
Nex , we plo he squa e o he end- o-end dis ance o he polyme o e ime (o numbe o
Mon e Ca lo sweeps). The end- o-end dis ance is simply he dis ance be ween he i s and las
monome on he chain: 𝑅𝑒=|𝑟𝑁−𝑟1|
We also plo he cumula i e mean o 𝑅𝑒
2, which means a each ime s ep we also plo he mean
o all p io alues o 𝑅𝑒
2 up o ha s ep. I is mo e na u al o analyze ⟨𝑅𝑒
2⟩ a he han ⟨𝑅𝑒⟩
because he squa ed dis ance a e ages in a s a is ically clean manne unde he mal luc ua ions
and adds linea ly ac oss independen segmen s, making i di ec ly compa able o analy ical
scaling laws.
Fo he ollowing plo , we ha e aken a e y la ge numbe o Mon e Ca lo s eps (1000000);
his ensu es he polyme has su icien ime o explo e i s con igu a ion space, and ha in u n
ensu es smoo he plo s wi h mo e s a is ical accu acy.
Figu e 7: Time se ies o squa ed end- o-end dis ance and i s mean, o a polyme chain o leng h 40
unde disc e e Rouse dynamics, 1000000 Mon e Ca lo s eps, 30X30 2D la ice, andom ini ial
con igu a ion
The cu e keeps luc ua ing. This is he beha io we obse ed when we we e isually
examining he di e en polyme con igu a ions on he la ice o e ime o a andom ini ial
con igu a ion – he polyme did no de ini i ely lean owa d ei he collapsing o swelling, bu
kep luc ua ing and explo ing di e en con o ma ions. In he abo e plo , we ha e plo ed he
squa e o he dis ance be ween he i s and las monome s as he polyme e ol es h ough
1000000 Mon e Ca lo s eps, and 𝑅𝑒
2 keeps luc ua ing as he polyme explo es di e en
con o ma ions. In sho , he polyme keeps wiggling o e e . We would expec i o spend equal
ime s e ched and comp essed – some hing which may no be isible di ec ly on he plo due
o a numbe o easons. Fi s , e en hough 1000000 s eps migh seem like a la ge enough
numbe , i may no always be su icien o exclude e ec s like he size o he box (la ice),
biases in he ini ial con igu a ion and slow mixing. To illus a e his, we un he same
simula ion as abo e once mo e, and see ha he cumula i e mean con e ges o a di e en alue
his ime.
Figu e 8: Re un o he abo e simula ion, same pa ame e s, same numbe o s eps, andom ini ial
con igu a ion
In he abo e plo , we isually see ha ⟨𝑅𝑒
2⟩ con e ges o a alue sligh ly abo e 250. Fo he
abo e simula ion, we ha e conside ed a sel -a oiding polyme wi h 𝑁=40 in wo dimensions,
(𝜈=3/4), hence we would expec ⟨𝑅𝑒
2⟩ ~ 𝑁2𝜈 ~ 𝑁3/2 ~ 253, which is indeed close o he
con e ging alue ha we isually see on he plo . In a di e en un o he same simula ion
( igu e 7), howe e , ⟨𝑅𝑒
2⟩ con e ges o a e y di e en (and much smalle ) alue. This a ia ion
a ises because each simula ion begins om a single, andomly gene a ed ini ial con igu a ion
and hus explo es only one s ochas ic ajec o y h ough he polyme ’s con igu a ion space.
Al hough long Mon e Ca lo uns allow he sys em o sample many con o ma ions, he sampling
emains limi ed o he subse accessible om ha pa icula ini ial con igu a ion poin wi hin a
ini e simula ion ime. In p inciple, ue ensemble con e gence would equi e a e aging o e
many independen ly ini ialized polyme s o unning ex emely long ajec o ies. Ob iously, in
o de o ob ain eliable es ima es o ⟨𝑅𝑒
2⟩ and i s cumula i e mean, i is essen ial ha he
obse a ion ime o he simula ion signi ican ly exceeds he polyme ’s cha ac e is ic
co ela ion ime, 𝜏𝑐∼𝑁2𝜈+1. Only beyond his imescale do successi e con igu a ions become
e ec i ely unco ela ed, allowing he unning mean o con e ge owa d he ue ensemble
a e age. This empo al deco ela ion equi emen is pa icula ly s ong o long polyme s, since
𝜏𝑐 g ows apidly wi h 𝑁.
Howe e , e en a e ensu ing long simula ion imes and a e aging o e mul iple independen
ealiza ions, we see ha he mean alue o ⟨𝑅𝑒
2⟩ does no necessa ily equal he simple scaling
𝑁2𝜈 ( igu es 9, 12). This is because he co ec ela ion is ⟨𝑅𝑒
2⟩=𝐾𝑎2𝑁2𝜈, whe e 𝐾 is a
p opo ionali y cons an ha depends on mic oscopic de ails o he model. The ac o 𝐾 is no
necessa ily 1; in la ice models i can be subs an ially smalle , leading o nume ically lowe
⟨𝑅𝑒
2⟩ alues han he ideal scaling es ima e.
When ⟨𝑅𝑒
2⟩ is plo ed agains 𝑁 on a log-log scale, we should gene ally ob ain a s aigh line,
and he slope p o ides an empi ical measu e o 2𝜈. In igu e 11, we ob ain a slope o abou 1.2
ins ead o he expec ed 1.5 (co esponding o 𝜈=3/4 in 2D). This de ia ion can possibly be
a ibu ed o incomple e elaxa ion, limi ed ensemble sampling, e c. In igu e 14, we c ea e he
same plo o six polyme s (wi h 𝑁=20,30,40,50,60,70), and we ge a sligh ly be e i ed
slope o abou 1.3.
Figu e 9: Re un o he abo e simula ion o longe ime, a e a e aging o e 10 independen uns,
same pa ame e s, same numbe o s eps, andom ini ial con igu a ion
Figu e 10: Time se ies o he cumula i e means o 𝑅𝑒
2 o 10 independen uns, and hei ensemble-
a e aged cu e
Figu e 11: log ⟨𝑅𝑒
2⟩- s-log𝑁 plo o he abo e simula ion
Figu e 12: Time se ies o 𝑅𝑒
2 and ⟨𝑅𝑒
2⟩ a e aged o e 100 independen uns
Figu e 13: 100 independen uns in g oups o 10, wi h e o bands a ound ensemble-a e aged cu e
Figu e 14: log ⟨𝑅𝑒
2⟩- s-log𝑁 plo o 6 polyme s
I would also be a good idea o look a he log ⟨𝑅𝑒
2⟩- s-log𝑁 plo o he abo e six polyme s
wi hou sel -a oidance; in o he wo ds, we assume all he abo e polyme s o be Gaussian his
ime. We could expec a slope o exac ly one, since o Gaussian polyme s ⟨𝑅𝑒
2⟩ ~ 𝑁, and his
ime we ge an almos pe ec ma ch ( igu e 15).
Figu e 15: log ⟨𝑅𝑒
2⟩- s-log𝑁 plo o 6 Gaussian polyme s
We now look a he p obabili y dis ibu ion 𝑃(𝑅𝑒) o he end- o-end dis ance o he o iginal
(non-Gaussian) polyme s ( igu e 16). Fo mode a e chain leng hs, he dis ibu ion is sligh ly
skewed, wi h a non-Gaussian ail a he han a pe ec symme ic bell shape. This asymme y
a ises because he polyme spends mos o i s ime in compac , coiled con igu a ions, while
occasionally explo ing a e s e ched con o ma ions ha ex end he end- o-end dis ance a
beyond he mean. Also, he e we ha e a e aged o e only eigh independen uns, hence
signi ican noise emains.
Figu e 16: P obabili y dis ibu ion o 𝑅𝑒 o h ee polyme s o leng hs 20, 40 and 80
We now look a plo s o 𝑅𝑒
2 and ⟨𝑅𝑒
2⟩ as unc ions o ime o a smalle polyme (𝑁=20) in a
la ge la ice (𝐿=50); his would elimina e e ec s o he la ice bounda ies on he mo ion o
he polyme , and he polyme would deco ela e as e . This ime, we plo o 500000 s eps o
p ope compa ison (1000000 s eps o 𝑁=40 would be equi alen o 500000 s eps o 𝑁=
20).
Figu e 17: Time se ies o 𝑅𝑒
2 and ⟨𝑅𝑒
2⟩ o a polyme wi h N=20, L=50, 500000 Mon e Ca lo s eps,
andom ini ial con igu a ion
Figu e 18: Re un o he abo e simula ion, same pa ame e s, same numbe o s eps, andom ini ial
con igu a ion
We clea ly see ha now he di e ences be ween he con e ging alue o ⟨𝑅𝑒
2⟩ o e e uns o
he code a e smalle . And ⟨𝑅𝑒
2⟩ does no con e ge o a alue nea 𝑁, bu always g ea e han 𝑁,
since we ha e included he e ec o olume exclusion in he abo e simula ions. We now plo
𝑅𝑒
2 and ⟨𝑅𝑒
2⟩ as unc ions o ime o a Gaussian polyme .
a e age o e mul iple independen samples. In igu e 26, we ha e plo ed he ensemble-
a e aged MSD cu e o he abo e polyme o e 10 independen uns. We see a cleane
c osso e om subdi usi e o di usi e egimes a 𝑡 ~ 102. Fo a la ge polyme (𝑁=40),
we see a slowe c osso e om subdi usi e o di usi e egime ( igu e 27), as expec ed om
𝐷𝑐𝑚 ~ 1/𝑁. In igu e 27, he c osso e om subdi usi e o di usi e egime akes place a
𝑡>103. This makes sense, because o his plo 𝑁=40, which means 𝑁2.5 =10119.29,
which is o he o de o 104 ( his is a sel -a oiding polyme ). We see a much dense sa u a ion
he e because his ime he polyme is signi ican ly long as compa ed o he la ice size (𝑁=
40, 𝐿=30).
Figu e 27: MSD- s- ime cu e o a longe polyme (N=40, L=30), 1000000 s eps, andom ini ial
con igu a ion
Figu e 28: MSD- s- ime cu e o a longe polyme (N=40, L=80) chain, 10000000 s eps, andom
ini ial con igu a ion
In igu e 28, we simula e a sel -a oiding polyme o he same leng h, bu in a la ge la ice and
o a much highe numbe o s eps (𝑁=40, 𝐿=80, 10000000 s eps). In his case oo, we see
simila beha io ( he o de o 𝜏𝑐 seems o be a ound 103), al hough he e is signi ican noise
and sa u a ion. The sa u a ion pe sis s e en in a bigge la ice and o e longe imes because
ou Mon e Ca lo scheme is local, and ou mo es elax he polyme ’s in e nal modes wi hou
e icien ly displacing i s cen e o mass.
Now, o compa e he beha io o polyme s wi h di e en leng hs, we escale he ime axis by
he cha ac e is ic elaxa ion ime o he chain, and plo MSD as a unc ion o 𝑡/𝑁2 ( igu e 29).
In he subdi usi e egime – whe e he monome mo ion ollows MSD ∼𝑡1/2 – o e laps well
o chains o di e en leng hs (conside ing his cu e has no been a e aged o e di e en
independen uns). Beyond he c osso e howe e , he long- ime di usi e egime (MSD ∼𝑡)
does no align pe ec ly unde his scaling. This misma ch a ises because a e deco ela ion,
he o e all cen e -o -mass di usion scales as 𝐷𝑐𝑚 ∼1/𝑁, meaning he scaled ime should
ins ead be 𝑡/𝑁, no 𝑡/𝑁2. In o he wo ds, he e a e wo dis inc dynamic scalings – 𝑡/𝑁2 o
subdi usi e in e nal elaxa ion and 𝑡/𝑁 o cen e -o -mass di usion – and a single plo canno
simul aneously escale bo h egimes in o pe ec o e lap.
Figu e 29: MSD- s-scaled ime ( /N2) cu e o Gaussian polyme s o di e en leng hs (N=10, 20, 30,
L=60), 500000 s eps, andom ini ial con igu a ion
The impo an poin is ha plo ing he MSD o di e en chain leng hs agains his scaled ime
𝑡/𝑁2 emo es he i ial shi in he c osso e posi ion ha would o he wise occu when using
unscaled ime 𝑡, whe e longe chains appea o elax much mo e slowly simply because hei
in insic imescale is la ge . O e all, his demons a es ha he unde lying elaxa ion
mechanism is he same ac oss di e en chain leng hs, and ha he appa en di e ences in
unscaled ime a ise solely om he 𝑁2 scaling o he global Rouse elaxa ion ime. In he plo ,
we also see ha he c osso e seems o be no exac ly a 𝑡/𝑁2∼1 (which we would ideally
expec ), bu a a sligh ly la ge alue. This shi is no su p ising, because we ha e implemen ed
a local Mon e Ca lo scheme – only local, dis ance-p ese ing mo es a e allowed – and hus
many a emp ed mo es esul in no displacemen a all. These mic oscopic de ails e ec i ely
mul iply he physical elaxa ion ime by a cons an 𝐾 (say), and he slowes mode eally elaxes
a 𝜏𝑐 ~ 𝐾𝑁2, leading o a c osso e a /N2 ~ 𝐾 a he han exac ly 1. Bu clea ly, a e
escaling by 𝑁2, he h ee cu es in he abo e plo collapse on o he same ajec o y and exhibi
he same subdi usi e 𝑡1/2 beha io and e en ual di usi e d i . Thus, he unde lying Rouse
scaling is cap u ed co ec ly. The no iceable noise in he high-𝑡 (di usi e) egion a ises om
la ge luc ua ions and limi ed s a is ics a long imes (independen con igu a ions a e ewe and
MSD accumula es s ochas ic a iance). To compensa e his, we nex plo he ensemble-
a e aged MSD (a e aged o e mul iple independen uns wi h di e en andom ini ial
con igu a ions), o polyme s o di e en leng hs ( igu es 30, 31, 32).
Figu e 30: Ensemble-a e aged MSD- s-scaled ime ( /N2) cu e o e 10 independen uns, o
Gaussian polyme s o leng hs N=10, 20, 30, L=60
Figu e 31: Ensemble-a e aged MSD- s-scaled ime ( /N2) cu e o e 10 independen uns, o
Gaussian polyme s o leng hs N=20, 40, 80, L=100
Figu e 32: Ensemble-a e aged MSD- s-scaled ime ( /N2) cu e o e 100 independen uns, o
Gaussian polyme s o leng hs N=20, 30, 40, 50, 60, 70, L=100
In he abo e plo s, he andom luc ua ions in he di usi e egion a e signi ican ly supp essed,
and we ge a smoo he and mo e s a is ically ep esen a i e depic ion o he MSD ac oss bo h
egimes.
Finally, we b ie ly examine he dynamics o he cen e o mass o a Gaussian polyme . By
acking he ins an aneous posi ion o he cen e o mass, 𝑟𝑐𝑚 =1
𝑁∑𝑟𝑖
𝑁
𝑖=1 , we can isualize how
he en i e polyme d i s as a single en i y o e ime. The plo ed ajec o y o he cen e o
mass ( igu e 33) o a polyme (𝑁=20) e eals a smoo h, wande ing pa h, eminiscen o a
wo-dimensional andom walk, con i ming ha al hough indi idual monome s unde go
complex co ela ed mo ions, hei collec i e mo ion beha es di usi ely on long imescales.
Figu e 33: T ajec o y o he cen e o mass o a Gaussian polyme wi h N=20, unde disc e e Rouse
dynamics
We now plo he ime se ies o he MSD o he cen e o mass, ⟨|𝑟𝑐𝑚(𝑡)−𝑟𝑐𝑚(0)|2⟩, o
polyme s o h ee di e en leng hs (𝑁=20,40,80), on he log-log scale ( igu e 34). All h ee
cu es show an almos pe ec linea dependence on ime, e lec ing he expec ed di usi e
scaling MSDcm ∼𝑡. The longe polyme s lie sys ema ically below he sho e ones, indica ing
a slowe di usion a e ( ecall 𝐷cm ∼1/𝑁). The cen e o mass o a longe polyme di uses
mo e sluggishly because i s o e all ic ion inc eases p opo ionally o he numbe o
monome s, while he he mal d i ing o ce emains cons an pe monome .
Figu e 34: Ensemble-a e aged MSD- s- ime cu es o e 10 independen uns, o Gaussian polyme s
o leng hs N=20, 40, 80, L=120
In he MSD plo , e y sho imes we e delibe a ely excluded. This is because a ea ly imes
he polyme ’s in e nal elaxa ion domina es, and he cen e o mass ba ely mo es. Remo ing
he e y ea ly ime poin s helps isola e he egime o s eady di usi e mo ion, whe e he cen e
o mass dynamics ha e deco ela ed om he ini ial con igu a ion and he expec ed 𝑡1 scaling
becomes clea and smoo h.
Figu e 35: Ensemble-a e aged MSD- s- ime cu es o e 10 independen uns, o sel -a oiding
polyme s o leng hs N=20, 40, 80, L=120
Fo a sel -a oiding polyme , he cen e o mass mo ion exhibi s slowe di usion ( igu e 35).
The sho es polyme (𝑁=20) seems o g ossly ollow he expec ed 𝑀𝑆𝐷∼𝑡 scaling;
howe e he e is no iceable la ening o longe polyme s, owing mainly o excluded olume
e ec s – monome s can now no longe eely pass h ough one ano he , and he longe chains
a e signi ican ly impac ed because o his; hey eel an inc eased e ec i e ic ion which causes
hem o di use mo e slowly, consis en wi h he scaling 𝐷cm ∼1/𝑁.
In conclusion, he ansi ion om a 𝑡1/2 scaling o a 𝑡 scaling in he mean squa ed displacemen
ma ks he c osso e om subdi usi e o di usi e beha io in he polyme ’s dynamics. In he
sho - ime egime (subdi usi e), each monome is apped by i s neighbo s along he chain,
and i s mo ion is hinde ed. A longe imes, once hese in e nal modes ha e elaxed su icien ly,
he en i e polyme mo es collec i ely and he mo ion becomes di usi e, domina ed by he ee
B ownian di usion o he polyme ’s cen e o mass.
Polyme Dynamics unde He e ogeneous Geome ic Rules
In his sec ion, we explo e he e ec o simple, bu spa ially he e ogeneous modi ica ions o
he geome ic ules on he beha io o he polyme . Unlike he s anda d Rouse dynamics we
ha e so a explo ed – whe e e e y monome obeys iden ical dis ance-p ese ing mo es and
ansla ional symme y is p ese ed along he chain – we now assign dis inc local ules o
al e na ing monome s. Apa om al e na ing copolyme s, we also build a oy model o a block
copolyme , whe e we di ide he polyme in o wo con inuous hal es and apply di e en
geome ic ules o each hal . The objec i e is no o eplica e any speci ic mic oscopic
mechanism, bu o explo e how nonuni o m mobili y, imposed pu ely h ough geome y and
bond-p ese ing la ice mo es, can quali a i ely model some aspec s o he nonequilib ium
polyme dynamics.
Based on wha we ha e explo ed so a , we aim o build an ex emely ligh weigh analogue o
si ua ions in which local polyme en i onmen s di e in empe a u e, e ec i e iscosi y,
c owding o o he mechanical cons ain s. In biological sys ems, polyme s such as DNA,
ch oma in ibe s and RNA egula ly encoun e he e ogeneous and spa ially a ying
su oundings: some egions encoun e s ong s e ic con inemen , o he s ansien ancho ing
and s ill o he s enhanced agi a ion. While such sys ems a e a mo e complex han any hing we
model he e, ou cons uc ions p o ide a simple way o in oduce local a iabili y wi hou
in oking explici o ces, po en ials o ene ge ic pa ame e s. All de ia ions om equilib ium
a ise solely om asymme ies in he allowed geome ic mo es o di e en monome s.
I is impo an o no e ha b eaking he uni o mi y o he geome ic ules b eaks he e ec i e
ansla ional symme y along he chain – e en i he polyme s a s in a ully symme ic
con o ma ion, he di e ing local mobili ies dis up de ailed balance a he algo i hmic le el,
o en pushing he sys em in o a d i en s eady s a e. In he o iginal Rouse model, e e y
monome ollows he same geome ic ules, has he same mobili y and a emp s he same kinds
o mo es equip obably. Thus, any mic oscopic change in con igu a ion has an equally p obable
e e se change. This one- o-one symme y be ween he p obabili y o o wa d and backwa d
ansi ions ensu es de ailed balance. E en hough he chain unde goes di usion, he e is no
buil -in p e e ence o mo ing in any pa icula di ec ion h ough he con igu a ion space.
Howe e in ou al e na ing- ule models, we will delibe a ely assign di e en kinds o mo es
o di e en monome s, and because o his, ce ain con igu a ion changes become mo e
p obable han hei exac e e ses. This b eaks de ailed balance, e en hough all bonds emain
in ac and no ex e nal o ces a e applied. The sys em no longe elaxes o he same equilib ium
s a is ics as he uni o m Rouse model, bu ins ead se les in o a nonequilib ium s eady s a e
d i en solely by di e ences in he pe mi ed geome ic mo ions along he chain.
To in oduce a simple o m o spa ial he e ogenei y in o he la ice polyme dynamics, we i s
y cons uc ing a oy nonequilib ium model in which di e en monome s obey di e en
geome ic upda e ules. The polyme is ini ialized as a sel -a oiding walk on a 40X40 la ice
( igu e 36), wi h all bond leng hs es ic ed o one la ice uni . We pe manen ly ix a small
ac ion o monome s (~ 3%) o hei ini ial la ice si e o mimic local en i onmen al pinning.
All emaining monome s mus p ese e he bond leng h o hei neighbo s a e e y s ep. The
al e na ing ules a e as ollows:
1. Monome s wi h e en-numbe indices ollow he amilia ule – hey can only a emp
mo es ha p ese e hei dis ances om hei nea es neighbo s.
2. Odd-indexed monome s a e allowed o a emp mo es o any o he eigh su ounding
la ice si es (including diagonal mo es) and a e addi ionally allowed o a emp wo-s ep
mo es chosen a andom, wi hin he same 8-neighbo geome y.
The inclusion o wo-s ep mo es in he ule se was in ended as a pu ely geome ic way o
encode he e ogeneous local mobili y, loosely analogous o segmen s o a chain expe iencing
egions o di e ing local iscosi y, c owding o agi a ion. Howe e , despi e his delibe a ely
he e ogeneous mechanism, he esul ing polyme dynamics is e y cons ained – he polyme
ha dly changes i s con igu a ion signi ican ly e en a e a la ge numbe o s eps.
The key eason is ha he la ge mo e se a ailable o he odd monome s is almos en i ely
supp essed by he bond-p ese a ion condi ion – which is a e y igh cons ain . Any candida e
mo e mus main ain a Manha an dis ance o exac ly one o bo h neighbo ing monome s, bu
diagonal mo es ypically inc ease his dis ance o wo o mo e, and wo-s ep mo es almos
always iola e he bond-leng h cons ain . Once olume exclusion and la ice bounda ies a e
also imposed, only a e y small subse o he nominally a ailable odd-monome mo es emain
iable. In p ac ice, e en hough odd monome s a emp mo e a ied mo es, almos all o hese
p oposals a e ejec ed.
Figu e 36: Mon e Ca lo simula ion o a sel -a oiding polyme chain o leng h 100 unde he ules o
ou al e na ing- ule oy model 1 on 40X40 2D la ice o andom ini ial con igu a ion
As a esul , he al e na ing- ule design in ou i s oy model does no gene a e he in ended
nonequilib ium e ec s, e en hough we s a om a andom ini ial con igu a ion. The chain is
essen ially “caged” by i s sel -a oiding s uc u e and he s ic bond-leng h equi emen , so he
he e ogenei y in he p oposal ules does no ansla e in o app eciable he e ogenei y in accep ed
mo es on he la ice. This highligh s an impo an lesson: unde s ong geome ic cons ain s,
blindly in oducing he e ogeneous ules and modi ying only he p oposal dis ibu ion is
insu icien o induce subs an ial dynamical di e ences.
I we emo e he condi ion o sel -a oidance om he abo e model, we see a e y sligh
inc ease in he a ia ions in he con igu a ion o he polyme h ough he same numbe o s eps
( igu e 37), bu his e ec is ex emely weak and he polyme s ill emains signi ican ly igid.
Figu e 37: Mon e Ca lo simula ion o a Gaussian polyme o leng h 100 unde he ules o ou
al e na ing- ule oy model 1 on 40X40 2D la ice o andom ini ial con igu a ion
Now, we y a di e en al e na ing- ule oy model. Fi s , we s a wi h a Gaussian polyme (no
sel -a oidance). Like be o e, e en-indexed monome s s ill ollow he amilia ule: a e e e y
mo e he nea es -neighbo dis ances mus be p ese ed. Bu his ime, o odd monome s we
do no check ha he nea es -neighbo dis ances emain cons an pos -mo e. This means, bonds
o neighbo s can s e ch o comp ess a bi a ily o odd-indexed monome s! Fu he , we assign
h ee possible ules o he odd-indexed monome s, which can be hough o ep esen egions
o he polyme expe iencing he e ogeneous o “ac i e’’ local en i onmen s, and a each s ep,
one ou o hese h ee ules is equip obably and andomly chosen and implemen ed on he odd-
indexed monome s. The h ee possible beha io s o he odd-indexed monome s a e:
1. They emain comple ely ixed o ha upda e a emp .
2. They mo e o one o he ou nea es la ice si es, o e lap allowed.
3. They mo e o any neighbo ing la ice si e ha lies one o wo s eps away (including
diagonal s eps).
These modes in en ionally di e in hei local dynamical eedom, and he andom swi ching
b eaks ull ansla ional and dynamical symme y along he chain. As in all p e ious
simula ions, he polyme is con ined by ha d bounda ies a he edges o he la ice.
Unde his second al e na ing- ule model, he mos s iking ea u e is he appea ance o la ge,
i egula and o en physically un ealis ic luc ua ions in he shape o he polyme ( igu e 38).
This beha io a ises p ima ily because odd-indexed monome s a e no longe equi ed o
main ain a ixed bond leng h wi h hei neighbo s, allowing he polyme o momen a ily
“s e ch’’ o “collapse’’ unp edic ably. E en hough he e en-indexed monome s s ill obey s ic
nea es -neighbo cons ain s – p o iding a pa ial sca old – he odd-indexed monome s
pe iodically b eak local s uc u e, p oducing ab up jumps, dis o ed segmen s and la ge
de ia ions om s anda d Rouse dynamics.
I is clea ha unde his model, he polyme keeps expanding o e all. As a consequence, he
squa ed gy a ion adius keeps s eadily inc easing ( igu e 39). As we ha e p e iously seen, in a
no mal Rouse o sel -a oiding chain, 𝑅𝑔
2(𝑡) luc ua es a ound an equilib ium alue, because
he dynamics sa is y de ailed balance. He e howe e , he luc ua ing and unbounded bond
leng hs o odd monome s explici ly b eak de ailed balance. The polyme is de ini i ely
“pushed’’ ou wa d by s e ches ha ha e no co esponding es o ing mo es. As a esul , 𝑅𝑔
2(𝑡)
inc eases s eadily a he han s abilizing, e lec ing a genuine nonequilib ium expansion d i en
by hese asymme ic geome ic ules.
Figu e 38: Mon e Ca lo simula ion o a Gaussian polyme o leng h 40 unde he ules o ou
al e na ing- ule oy model 2 on 80X80 2D la ice o andom ini ial con igu a ion
Figu e 39: Time se ies o gy a ion adius squa ed o a Gaussian polyme o leng h 40 unde he ules
o ou al e na ing- ule oy model 2 on 80X80 2D la ice o andom ini ial con igu a ion
One may ask how luc ua ing bond leng hs a e e en possible in his model, gi en ha e e y
odd monome is di ec ly connec ed o wo e en monome s whose bond leng hs a e s ic ly
In igu e 46, we examine he MSD o bo h he d i en and cons ained blocks o he polyme
sepa a ely. We plo he unning mean o all he MSDs o all he monome s wi hin each hal
(o ange: mean o e i s 25; blue: mean o e las 25), aken ela i e o he un’s ini ial posi ions.
A e aging ac oss he whole hal educes single-si e noise and highligh s he sys ema ic
mobili y di e ence be ween he blocks – he o ange mean MSD ises mo e s eeply because he
d i en monome s ha e la ge accessible mo e se s and highe a emp equency as compa ed
o he cons ained (blue) monome s. Plo ing he MSD o a single “ ep esen a i e” monome
om each block can gi e ambiguous o misleading signals ( o ins ance, i he chosen monome
is e y close o he ends).
Figu e 46: MSD- s- ime plo s o bo h d i en and cons ained blocks o he block copolyme ,
a e aged o e all monome s in each hal
The shaded anslucen egions a ound he cu es ep esen he s anda d e o o he mean
ac oss hose six independen uns. We see ha he d i en hal genuinely explo es mo e
con igu a ion space (so i s a e aged MSD is la ge ), bu global geome y and exclusion p e en
unbounded swelling (𝑅𝑔
2 he e o e luc ua es a ound a ini e ange). This model is a minimal,
geome ic oy ha cap u es he in ui ion o a localized mobili y/ empe a u e g adien : s onge
local ac i i y ( he d i en block) inc eases local luc ua ions and di usion, ye connec i i y and
excluded- olume couple hose luc ua ions o he es o he chain and keep he whole polyme
in a cons ained, quasi-s eady s a e.
I is clea ha he MSD cu es o he o ange and blue monome s clea ly di e ge: he d i en
block exhibi s a much s eepe g ow h in MSD (as compa ed o he cons ained block),
e lec ing i s highe local mobili y and la ge accessible con igu a ional olume. I is in e es ing
o ocus on wha happens a he in e ace – he bond be ween he las o ange and i s blue
monome s. The las o ange monome is ee o a emp all he long- ange “d i en” mo es like
he o he o ange monome s, bu each mo e mus s ill main ain a alid bond leng h wi h i s
immedia e blue neighbo ( alid acco ding o he ules ollowed by he blue monome s), so he
blue block en o ces s ic e geome ic cons ain s on he las o ange monome ha limi which
o hose mo es ac ually ge accep ed. In ou Mon e Ca lo scheme, bond alidi y is checked
only locally; each ime a mo e is a emp ed on a pa icula monome , he cons ain s on he
monome a e checked only agains i s wo immedia e neighbo s. Thus es ic ions a he
in e ace p opaga e only one bond deep, and do no globally supp ess mobili y ac oss he en i e
o ange block. The es o he o ange segmen emains highly mobile because once a mo e
sa is ies local geome y, i incu s no penal y om dis an cons ain s. In igu e 47, he black
cu e acks he MSD o he las o ange monome ( he in e ace o ange monome di ec ly
bonded o he i s blue monome ). As expec ed, i s di usion lies in be ween he ully d i en
o ange block and he cons ained blue block.
Figu e 47: MSD- s- ime plo s o he d i en block, he cons ained blocks (a e aged o e all
monome s in each hal ) and he las d i en monome a he in e ace o he block copolyme , a e aged
o e 100 independen uns
Figu e 48: MSD o he cen e o mass o he block copolyme
One migh expec ha o e long imes he en i e polyme di uses as a single objec , e en ually
causing he MSD o all monome s (o ange as well as blue) o con e ge. Howe e in his model,
he di e ing dynamical ules a e applied con inuously, no jus a ini ializa ion, and he d i en
and cons ained blocks emain kine ically dis inc a all imes. This is why he blue MSD cu e
ne e me ges wi h he o ange cu e, e en a e long imes, and ou copolyme canno be a ed
as a single B ownian pa icle (i is a composi e objec wi h a sus ained mobili y g adien ). While
he cen e o mass o he en i e polyme unde goes di usion go e ned by all monome s
collec i ely (see igu e 48), he in e nal ela i e mo ion con inues o e lec he imposed
he e ogeneous dynamics. Thus, he blue block emains sys ema ically slowe , and i s MSD
cu e does no asymp o ically me ge wi h he o ange MSD. This dynamical asymme y is no
some hing ha “a e ages ou ”, because i is ac i ely en o ced a e e y upda e s ep.
In igu e 49 ( he highe noise is because he e we ha e no a e aged o e 100 independen uns),
we emo e he asymme y in he a emp - a e; o ange monome s a e no longe chosen o upda e
wice as o en as he blue ones, now all monome s a e selec ed o upda es wi h equal
p obabili y. F om he posi ion o he black cu e, we see ha now he las o ange monome a
he in e ace almos does no exhibi he in e media e di usi i y obse ed ea lie . Al hough i
s ill has he la ge geome ic mo e se cha ac e is ic o he d i en block, mos o hose mo es
a e ejec ed because i is now no chosen mo e equen ly han he blue monome s, and as
be o e, he bond o i s blue neighbo mus emain wi hin he cons ained se o allowed
dis ances. Wi hou being upda ed mo e equen ly, he in e ace bead becomes limi ed by he
mobili y o he slowe block i is e he ed o, and i s ex a eedom is ne e ully exp essed in
he dynamics. Consequen ly, i s MSD lies much close o he blue cu e a he han
in e pola ing be ween o ange and blue as in igu e 47.
Figu e 49: MSD- s- ime plo s o he d i en block, he cons ained blocks (a e aged o e all
monome s in each hal ) and he las d i en monome a he in e ace o he block copolyme , bo h
o ange and blue monome s now equally likely o be chosen a each s ep
In a physical o biological con ex , such a he e ogeneous block copolyme could co espond o
a chain embedded in an inhomogeneous en i onmen , o example one expe iencing a g adual
empe a u e g adien o a spa ial a ia ion in local c owding o sol en densi y om i s one end
o he o he . The d i en block would hen ep esen a “ho e ” and less c owded egion whe e
he mal agi a ion is s onge and monome mo ion is enhanced, while he cons ained block
would co espond o a “colde ,” mo e iscous o s e ically hinde ed egion. Simila si ua ions
occu in li ing cells, whe e biopolyme s can span egions wi h signi ican ly di e en
mic oen i onmen s. Thus, his oy model – despi e being pu ely geome ic – cap u es an
essen ial idea: how local he e ogenei y in e ec i e empe a u e o c owding can shape global
polyme con o ma ion and dynamics, p oducing nonequilib ium s eady s a es ha s ill espec
s ong geome ic cons ain s.
Figu e 50: Snapsho om a di e en un, wi h sligh ly di e en pa ame e s, o he block copolyme
model discussed abo e
In he end, i is impo an o no e ha all he models explo ed in his sec ion a e no in ended
o mimic any speci ic biological mechanism, bu hey p o ide good illus a ions o how
he e ogeneous local mobili ies can gene a e nonequilib ium beha io e en in an o he wise
s uc u eless polyme . I would be in e es ing in u u e wo k o es addi ional combina ions
and a ia ions o he e ogenous geome ic ules and cons ain s. I is also c ucial o no e ha he
esul ing dynamics in ou al e na e- ule model do no gua an ee inc eased luc ua ions in
gene al; depending on he speci ic choice o he ules, some po ions o he chain may exhibi
la ge con igu a ion-space excu sions while o he po ions may become mo e cons ained (as
compa ed o s anda d Rouse-like dynamics). This lexibili y could be exploi ed o p obe a
ela i ely wide spec um o beha io – om unexpec edly s i ened mo ion o ex emely quick
elaxa ion – wi hin a single uni ying geome ic amewo k. O e all, hese oy models p o ide
a con olled en i onmen o examining how spa ial he e ogenei y alone, implemen ed h ough
simple la ice-based ules, can p oduce depa u es om he equilib ium-like beha io . Bu o
cou se, hese a e oy models and a e no in ended o p o ide quan i a i e p edic ions o eal,
nonequilib ium polyme s.
Discussion and Conclusion
In his epo , we ha e explo ed Rouse model analy ically, and hen simula ed a simpli ied,
disc e e e sion o he Rouse model wi h sel -a oidance in o de o unde s and how global
polyme beha io can eme ge om simple local ules. By cons aining each monome o mo e
in ways ha p ese e he nea es -neighbo dis ances and a oid o e lap, we ha e obse ed how
a polyme chain e ol es on a 2D la ice h ough pu ely en opic dynamics. Despi e he absence
o explici ene ge ic e ms o sol en in e ac ions, he sys em displayed key quali a i e ea u es
o eal polyme mo ion – elaxa ion om s e ched o andom ini ial con igu a ions,
luc ua ions in end- o-end dis ance and gy a ion adius, and a clea c osso e om subdi usi e
(𝑡1/2) o di usi e (𝑡) egimes o e ime. These esul s highligh how he in e play o
connec i i y and sel -a oidance alone can encode much o he essen ial physics o polyme
mo ion.
Then we in oduced he e ogenei y in o he upda e ules o p obe he onse o nonequilib ium
beha io in a con olled manne . The al e na ing- ule models and he block copolyme model
demons a ed ha modi ying only he p oposal mechanisms – while keeping all mo es pu ely
geome ic – can b eak de ailed balance and p oduce a wide ange o in e es ing beha io . Some
he e ogeneous models p oduced minimal change in global beha io , because he ixed bond-
leng h and sel -a oidance cons ain s ejec mos asymme ic p oposals. Howe e , in models
in which bond leng h p ese a ion was elaxed o a subse o monome s, we saw s ong
nonequilib ium e ec s, including pe sis en expansion and luc ua ions in he gy a ion adius.
In e media e models combining enhanced mobili y, in e mi en immobiliza ion and geome ic
cons ain s o ixed bond leng h and sel -a oidance p oduced nonequilib ium s eady s a es ha
emained close o equilib ium, wi h he polyme “b ea hing’’ a ound a bounded size a he han
d i ing ou wa d inde ini ely.
In eal sys ems, polyme con o ma ions a ise om a compe i ion be ween ene gy and en opy:
a ac i e in e ac ions be ween monome s in poo sol en s (sol en s ha epel monome s)
cause he chain o collapse, while he mal mo ion in good sol en s (sol en s ha a ac
monome s) p omo es swelling. Al hough he models we ha e explo ed in his epo do no
include explici a ac ion o epulsion, he geome ic cons ain o sel -a oidance encode a
beha io simila o en opic esis ance o collapse in he dynamics. Ul ima ely, hese esul s
illus a e ha e en a minimal ule se – pu ely local, geome ic and some imes he e ogeneous
– can gi e ise o collec i e, eme gen dynamics a he mac oscopic le el. This e lec s one o
he cen al insigh s in he s udy o complex sys ems: global beha io o en s ems no om
in insically complex ules o in ica e mic oscopic laws, bu om he epea ed logic o simple,
local cons ain s.
Re e ences
• Schiessel, H. (2014). Biophysics o Beginne s: A Jou ney h ough he Cell Nucleus. Pan
S an o d Publishing.
• Padding, J. T. (2005). Theo y o Polyme Dynamics. Uni e si y o Camb idge.
• Li, B., Mad as, N., & Sokal, A. (1994). C i ical Exponen s, Hype scaling and Uni e sal
Ampli ude Ra ios o Two- and Th ee-Dimensional Sel -A oiding Walks. a Xi :hep-
la /9409003.
• Van Leeuwen, J., & D zewiński, A. (2009). S ochas ic La ice Models o he Dynamics o
Linea Polyme s. Physics Repo s, 475(5–6), 53–90.
• Newman, M. E. J., & Ba kema, G. T. (1999). Mon e Ca lo Me hods in S a is ical Physics.
Ox o d Uni e si y P ess.
Appendix: End-Monome In e ac ion as a Minimal Model o Polyme
Collapse
In his appendix, we explo e a e y simpli ied polyme model, whe e we explici ly in oduce
ene ge ic in e ac ions, bu only be ween he wo end monome s (all in e nal monome s eel no
ene ge ic bias a all). This minimal se up le s us p obe how global con o ma ional ansi ions
ge modi ied by e en jus bounda y in e ac ions. So e e y in e nal monome does no
expe ience any ene ge ic bias, and in con ex o ou p e ious discussions, s ill unde goes pu ely
geome ic Rouse-like mo es. The e is an (a ac i e o epulsi e) in e ac ion only a he end
poin s (be ween he wo ends o he chain). As he in e ac ion s eng h 𝐽 a ies, he polyme
can nea ly ansi ion om an ex ended, swollen s a e in o a collapsed con igu a ion whe e he
ends p e e o mee . In a sense, he polyme ’s long- e m beha io is la gely de e mined by he
end monome s, and he es o he chain nego ia es wi h wha e e he ends demand. To
o malize his, we w i e a Hamil onian in ol ing only he end monome s (we use a K onecke
del a con ac e m be ween he wo end monome s indexed 1 and 𝑁):
𝐻=−𝐽𝛿𝑟1,𝑟𝑁
whe e 𝑟1 and 𝑟𝑁 deno e he la ice posi ions o he i s and las monome . Acco ding o he
de ini ion o he K onecke del a, 𝛿𝑟1,𝑟𝑁=1 i 𝑟1=𝑟𝑁 ( he wo end monome s occupy he
same la ice si e), 𝛿𝑟1,𝑟𝑁=0 o he wise. This means when he end monome s in e ac , he
Hamil onian is 𝐻=−𝐽, and i 𝐽>0, we see ha he e is end- o-end in e ac ions would be
ene ge ically a o ed, since hey lowe he o e all ene gy o he chain by −𝐽. (Th oughou his
appendix, we assume 𝐽>0. I we wan he end monome s o epel, we can penalize end- o-
end in e ac ion by se ing 𝐽<0.) Thus in a way, he en i e ene gy o he polyme comes om
a single bina y e en : ei he he ends mee (and he sys em gains an ene gy −𝐽), o hey do no
(and he Hamil onian is ze o). No in e nal monome has any pai wise o sel -in e ac ion e m.
Despi e i s simplici y, his Hamil onian ep esen s a clea physical compe i ion be ween
en opy and ene gy. A 𝐽=0, he polyme beha es jus like an ideal 2D andom walk, wi h no
eason o he ends o a ac each o he . In ha case, he p obabili y ha he chain
spon aneously o ms a loop scales as:
𝑝0 ~ 1
𝜋𝑁
In ui i ely, we can easily see why he abo e scaling makes sense o an ideal, Gaussian polyme
in 2D. We ha e al eady looked a he analy ics o a 2D andom walk. Assume he walk s a s
a a ixed poin ( he ixed end). Then he posi ion o he ee end a e 𝑁 s eps can be hough
o as he endpoin o he 2D andom walk. As we ha e seen, a e 𝑁 s eps he ypical end- o-
end dis ance scales as Δ𝑟 ~ √⟨Δ𝑟2〉∝√𝑁
(because ⟨Δ𝑟2〉=𝑎2𝑁, assume 𝑎=1 o
simplici y). Now we can hink o he endpoin as being dis ibu ed o e a oughly ci cula
egion wi h a adius o he o de o √𝑁. O cou se, his ci cula -cloud pic u e is no comple ely
accu a e, and hence his is no a igo ous de i a ion. We a e simply in e es ed in an in ui i e
jus i ica ion o why he p obabili y o he end monome s in e ac ing and o ming a loop
spon aneously (𝐽=0) scales as 1/𝜋𝑁 ( o a polyme wi h 𝑁 monome s).
So i he ypical dis ance scale is √𝑁, hen he cha ac e is ic egion explo ed by he endpoin
has an a ea on he o de o 𝜋(√𝑁)2=𝜋𝑁 (assuming a ci cula egion, o cou se). This a ea
ep esen s he “ a ge zone’’ in which he endpoin is mos likely o be ound, and al hough o
cou se he exac densi y is no necessa ily uni o m, o he pu pose o ou es ima e, he endpoin
may be ega ded as being sp ead oughly e enly o e his egion. On he 2D la ice, a single
si e occupies an a ea o one uni cell, and hence he p obabili y ha he ee end o he polyme
lands p ecisely on he speci ic la ice si e occupied by he i s monome ( he o he end) is
app oxima ely he a io o he a ea o one la ice cell o he o al a ea o he endpoin cloud:
𝑝0 ~ 1
𝜋𝑁
Clea ly, he e u n p obabili y dec eases in e sely wi h 𝑁: o longe polyme s, he endpoin
wande s o e a co espondingly la ge egion, and he chance ha i e u ns exac ly o he o igin
becomes p opo ionally smalle . E en o 𝑁=40, 𝑝0 ~ 1/𝜋𝑁≈0.008. Hence, wi hou
explici ene ge ic in e ac ions, i is e y unlikely ha he end monome s in e ac and he
polyme o ms a loop spon aneously. We now explo e wha happens when 𝐽≠0.
Ini ially, we ied o di ec ly model he collapse o he polyme using ou local Mon e Ca lo
scheme on he 2D la ice, in o de o isualize how he no malized end- o-end squa ed dis ance,
⟨𝑅𝑒
2⟩/𝑁, a ies wi h he in e ac ion s eng h 𝐽. When 𝐽 is e y small, we would no expec he
polyme o collapse, and ⟨𝑅𝑒
2⟩/𝑁 would no a y signi ican ly (since i is he no malized mean-
squa ed end- o-end dis ance). We would hus expec a pla eau on he ⟨𝑅𝑒
2⟩/𝑁- s-𝐽 plo o small
𝐽. As we keep inc easing 𝐽, we would expec a ela i ely sha p d op owa d ze o (once he
polyme decides o o m a loop, he end- o-end dis ance becomes ze o). Also, o longe
polyme s (la ge 𝑁), his ansi ion should occu a la ge 𝐽 (because he en opic penal y o
b inging wo dis an ends oge he inc eases wi h chain leng h).
Howe e , he Mon e Ca lo simula ions did no ep oduce his expec ed beha io ( igu e 51).
Despi e ex ensi e sampling and a e aging o e independen uns, he plo s o ⟨𝑅𝑒
2⟩/𝑁- s-𝐽 o
h ee polyme s (o di e en leng hs) we e nea ly la on a e age.
Figu e 51: ⟨𝑅𝑒
2⟩/𝑁- s-𝐽 plo s using Mon e Ca lo simula ions (a e aged o e 2 independen uns,
e o ba s plo ed) o h ee polyme s o leng hs 20, 40 and 80, wi h end-only a ac ion
The eason is because he Hamil onian 𝐻=−𝐽𝛿𝑟1,𝑟𝑁 only a ec s he wo end monome s. Fo
mos polyme con igu a ions, his con ac e en is ex emely a e (𝑝0∼1/𝜋𝑁). E en o
mode a e chain leng hs, 𝑝0 is al eady so small ha he e ec i e empe a u e ange explo ed by
e en 𝐽≤500 is a oo weak o o e come he en opic cos . Hence, he abo e simula ion did
no show a collapse ansi ion. To see ha di ec ly, we would need ex emely la ge 𝐽 alues
plus ex emely long simula ion imes, which is compu a ionally expensi e and ime-
consuming.
To imp o e explo a ion o con o ma ions, we nex inco po a ed pi o mo es, whe e a
andomly chosen monome ac s as a hinge and he emaining chain segmen is o a ed o
e lec ed by a la ice symme y ope a ion. Pi o mo es do no al e bond leng hs o iola e
chain connec i i y, bu hey can d ama ically change global s uc u e in a single s ep, and
inc ease he p obabili y o he end monome s mee ing. Despi e his, and despi e longe uns
o e mul iple independen his o ies, and ela i ely sho polyme s (𝑁=20), he end- o-end
con ac emained e ec i ely unobse ed unde ou Mon e Ca lo scheme. Again, his is because
he Hamil onian con ains only a K onecke -del a in e ac ion localized a he wo ends, and he
ene ge ic gain occu s only when hose ends p ecisely occupy he same la ice si e. Since his
ep esen s a single mic os a e ou o an exponen ially la ge con igu a ion space, unbiased
Mon e Ca lo a ely encoun e s i .
In o de o cap u e he co ec quali a i e physics wi hou such compu a ional expense, we
analy ically simpli ied he sys em in o a wo-s a e model. The polyme is ea ed as ei he “non-
con ac ,” whe e we assume he ideal esul ⟨𝑅𝑒
2⟩≈𝑁; o “con ac ,” whe e he wo ends o e lap
and 𝑅𝑒
2≈0. The ela i e s a is ical weigh o hese wo s a es is go e ned by he Bol zmann
ac o associa ed wi h he con ac ene gy −𝐽. I 𝑝0 is he con ac p obabili y a 𝐽=0, hen he
con ac p obabili y a a bi a y 𝐽 becomes:
𝑝𝐽=𝑝0𝑒𝛽𝐽
(1−𝑝0)+𝑝0𝑒𝛽𝐽
The ensemble-a e aged end- o-end dis ance is hen app oxima ed as:
⟨𝑅𝑒
2⟩=𝑁(1−𝑝𝐽)
This o mula cap u es he co ec physical end. Fo low 𝐽 (when 𝐽→0), 𝑒𝛽𝐽 →1 and 𝑝𝐽→
𝑝0. Since 𝑝0 is usually negligibly small (e en o mode a e leng h polyme s), o low 𝐽 we
e ec i ely ha e 𝑝𝐽→0. Since he con ac p obabili y is anishingly small a low 𝐽, om
⟨𝑅𝑒
2⟩=𝑁(1−𝑝𝐽) we ha e ⟨𝑅𝑒
2⟩→𝑁, and hence ⟨𝑅𝑒
2⟩/𝑁→1 (ex ended s a e, en opy
domina es!). Fo la ge 𝐽 (when e ec i ely 𝐽→∞), 𝑝0𝑒𝛽𝐽 →∞, and hence (1−𝑝0)+𝑝0𝑒𝛽𝐽 ≈
𝑝0𝑒𝛽𝐽, which means 𝑝𝐽→1. Combining his wi h ⟨𝑅𝑒
2⟩=𝑁(1−𝑝𝐽) gi es us ⟨𝑅𝑒
2⟩/𝑁→0
(collapsed s a e, ene gy wins!). We now plo ⟨𝑅𝑒
2⟩/𝑁- s-𝐽 using he educed wo-s a e model,
o he same h ee polyme s ( igu e 52).
We see ha he plo o ⟨𝑅𝑒
2⟩/𝑁- s-𝐽 cleanly displays he expec ed c osso e om a swollen,
coil-like con igu a ion cha ac e ized by ⟨𝑅𝑒
2⟩/𝑁≈1, o a compac , looped s a e cha ac e ized
by ⟨𝑅𝑒
2⟩/𝑁≈0. Fo ela i ely small alues o 𝐽, he cu e emains la o a wide ange,
e lec ing pu ely en opic beha io . Then he cu e d ops ab up ly o e a ela i ely na ow
in e al, indica ing a s ongly coope a i e ansi ion (which esembles a i s -o de phase
ansi ion). The abo e plo is ac ually a semilog plo , because we ha e plo ed 𝐽 on a loga i hmic
axis o be e isualiza ion ( he collapsed s a e ex ends o e se e al o de s o magni ude in 𝐽,
and using a linea scale would comp ess he en i e ansi ion in o an in isible sli e ).
Figu e 52: ⟨𝑅𝑒
2⟩/𝑁- s-𝐽 plo s o h ee polyme s (N=20,40,80) wi h end-only a ac ion, using he
simpli ied wo-s a e analy ic model
I is impo an o no e ha he e is no ue i s -o de phase ansi ion in ou model, e en i he
ansi ion appea s i s -o de -like. This is simply because o a ue he modynamic i s -o de
ansi ion, he obse able should become non-analy ic in some ex eme limi ( o example,
𝑁→∞). He e, howe e , he polyme is a ini e chain wi h only a single ene ge ic in e ac ion
be ween i s wo end monome s; no ma e how sha ply he end- o-end dis ance changes, he
ee ene gy emains smoo h and analy ic o all ini e 𝑁.
This s eep sigmoidal p o ile also esembles he beha io o molecula swi ches o en exploi ed
in biological sys ems, o ins ance, a small change in binding ene gy can lip an en i e polyme
be ween dis inc s uc u al s a es. The pa ame e 𝐽 he e ac s as an e ec i e con ol knob, uning
he compe i ion be ween en opy and end-only a ac ion and he eby igge ing a swi ch-like
collapse.
Now, we look a plo s o he con ac p obabili y 𝑝𝐽 as unc ions o he in e ac ion s eng h 𝐽,
o he h ee polyme s as be o e ( igu e 53). The 𝑝𝐽- s-𝐽 plo shows he co esponding g ow h
o he end- o-end con ac p obabili y as he in e ac ion s eng h inc eases. Fo small 𝐽, 𝑝𝐽
emains close o ze o because he en opic cos o b inging he wo ends oge he domina es;
once 𝐽 app oaches he c osso e poin , he p obabili y ises sha ply in a sigmoidal, swi ch-like
manne . This beha io is ully consis en wi h he ⟨𝑅𝑒
2⟩/𝑁 cu es: he collapse o he polyme
coincides wi h he apid inc ease in he likelihood o end- o-end binding.
