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Polymer Statistics III Mixtures of polymers
applications to proteins and nucleic acids
recap: scaling exponents and coil-to-globule
-Tltr(Lt1p»lttLrtics
rt ix iÀG
Cotsisr oT fdo oQ iadtr biffieaatq- ctenìcAlHilrru n-l5?eei€3 .
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A,urcÈ +lono Gl:-rnlrrlos Ix&n?G.: A ?au(nAt ;d fuo» bul€§T
lf rrt€ nixrr)ttf Ca'a'i579 o; *e:Ùer?AL DtFrett?^'y; ÈtAges
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-Tltr(Lt1p»lttLrtics
rt ix iÀG
Cotsisr oT fdo oQ iadtr biffieaatq- ctenìcAlHilrru n-l5?eei€3 .
tfruEmìXnnrìsu^JìroonAd»Aucsn?oilarrSA/tEil-rgnnixaD o rJ ft flo urctt*tl go*(E r rtlf- nì xa11ttÉ i s
A,urcÈ +lono Gl:-rnlrrlos Ix&n?G.: A ?au(nAt ;d fuo» bul€§T
lf rrt€ nixrr)ttf Ca'a'i579 o; *e:Ùer?AL DtFrett?^'y; ÈtAges
( nrAoSS W;--' tl » irevr'alt cen?eQ'no"r:) ' i' i 5 64 aiaN
+leT€le6gzJeouS IoZ*n?É, i oit- AÀb \^'ArElt
-Tltr(Lt1p»lttLrtics
rt ix iÀG
Cotsisr oT fdo oQ iadtr biffieaatq- ctenìcAlHilrru n-l5?eei€3 .
tfruEmìXnnrìsu^JìroonAd»Aucsn?oilarrSA/tEil-rgnnixaD o rJ ft flo urctt*tl go*(E r rtlf- nì xa11ttÉ i s
A,urcÈ +lono Gl:-rnlrrlos Ix&n?G.: A ?au(nAt ;d fuo» bul€§T
lf rrt€ nixrr)ttf Ca'a'i579 o; *e:Ùer?AL DtFrett?^'y; ÈtAges
( nrAoSS W;--' tl » irevr'alt cen?eQ'no"r:) ' i' i 5 64 aiaN
+leT€le6gzJeouS IoZ*n?É, i oit- AÀb \^'ArElt
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V6 Vo -+ Va+Vs
, V oLùrW r(wcrt'oi ;
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, V oLùrW r(wcrt'oi ;
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w, Ass orl, t*AT hw auz c-orlpo^JÉa,rs a^ tsa xscABrD
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V6 Vo -+ Va+Vs
, V oLùrW r(wcrt'oi ;
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w, Ass orl, t*AT hw auz c-orlpo^JÉa,rs a^ tsa xscABrD
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V6 Vo -+ Va+Vs
, V oLùrW r(wcrt'oi ;
rV*VaP=-=Tft Vror Va + Vs Vtrt V* + Vg
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w, Ass orl, t*AT hw auz c-orlpo^JÉa,rs a^ tsa xscABrD
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b
Degree of polymerization
----|.e-^t-, nrrv [, 5= (a 0d -Q-ò \
^rn'ce ofoaàùlt*noJs
/Vl = ru0rtBuR. a€ LfrTiC€ =ìtz:-NO, Ng : D?6nE€ or ?ocvnaz^?4r,o^J
. lnl A qono &,t(n1g mtrrr.lnr 6 ft Adb 6 eActt ttotzci(E
H[S
-Q-Ag = lw
' a$é u?€è;4 À b#p1 n ixrJ6 ('* A ?ani "A' "^,=)
-§LA = rnf.
, Tt(f c+tA,JW ì^l gr'rrno?( is &'Ùe?ù Bt :
AS^ = sJ* rcrid& sf* niri"rc -
= Ks \-Q*o Kg 4-0+ =
=&!dLr ò?*AcuA'rS losiiwA*<L
= Kb -L\$r = Ka lq Étr ò _rLA 'r Ò /40
----|.e-^t-, nrrv [, 5= (a 0d -Q-ò \
^rn'ce ofoaàùlt*noJs
/Vl = ru0rtBuR. a€ LfrTiC€ =ìtz:-NO, Ng : D?6nE€ or ?ocvnaz^?4r,o^J
. lnl A qono &,t(n1g mtrrr.lnr 6 ft Adb 6 eActt ttotzci(E
H[S
-Q-Ag = lw
' a$é u?€è;4 À b#p1 n ixrJ6 ('* A ?ani "A' "^,=)
-§LA = rnf.
, Tt(f c+tA,JW ì^l gr'rrno?( is &'Ùe?ù Bt :
AS^ = sJ* rcrid& sf* niri"rc -
= Ks \-Q*o Kg 4-0+ =
=&!dLr ò?*AcuA'rS losiiwA*<L
= Kb -L\$r = Ka lq Étr ò _rLA 'r Ò /40
----|.e-^t-, nrrv [, 5= (a 0d -Q-ò \
^rn'ce ofoaàùlt*noJs
/Vl = ru0rtBuR. a€ LfrTiC€ =ìtz:-NO, Ng : D?6nE€ or ?ocvnaz^?4r,o^J
. lnl A qono &,t(n1g mtrrr.lnr 6 ft Adb 6 eActt ttotzci(E
H[S
-Q-Ag = lw
' a$é u?€è;4 À b#p1 n ixrJ6 ('* A ?ani "A' "^,=)
-§LA = rnf.
, Tt(f c+tA,JW ì^l gr'rrno?( is &'Ùe?ù Bt :
AS^ = sJ* rcrid& sf* niri"rc -
= Ks \-Q*o Kg 4-0+ =
=&!dLr ò?*AcuA'rS losiiwA*<L
= Kb -L\$r = Ka lq Étr ò _rLA 'r Ò /40
----|.e-^t-, nrrv [, 5= (a 0d -Q-ò \
^rn'ce ofoaàùlt*noJs
/Vl = ru0rtBuR. a€ LfrTiC€ =ìtz:-NO, Ng : D?6nE€ or ?ocvnaz^?4r,o^J
. lnl A qono &,t(n1g mtrrr.lnr 6 ft Adb 6 eActt ttotzci(E
H[S
-Q-Ag = lw
' a$é u?€è;4 À b#p1 n ixrJ6 ('* A ?ani "A' "^,=)
-§LA = rnf.
, Tt(f c+tA,JW ì^l gr'rrno?( is &'Ùe?ù Bt :
AS^ = sJ* rcrid& sf* niri"rc -
= Ks \-Q*o Kg 4-0+ =
=&!dLr ò?*AcuA'rS losiiwA*<L
= Kb -L\$r = Ka lq Étr ò _rLA 'r Ò /40
ASn^^ixm
= /YtA ASo r
= € K6
YAS,=/\
lù-l-L nro
44r+ k4l{Ng
fdor*uuS.}pdrto")S
c{4Ad(rS
, ?aLL( r:a.rL
AlmoP< foftbulrio"lS / AÀ»
QdCrÙtA,tL
(otvnZrz Ègl»S-
O
per lattice site
total entropy
ASn^^ixm
= /YtA ASo r
= € K6
YAS,=/\
lù-l-L nro
44r+ k4l{Ng
fdor*uuS.}pdrto")S
c{4Ad(rS
, ?aLL( r:a.rL
AlmoP< foftbulrio"lS / AÀ»
QdCrÙtA,tL
(otvnZrz Ègl»S-
O
ASn^^ixm
= /YtA ASo r
= € K6
YAS,=/\
lù-l-L nro
44r+ k4l{Ng
fdor*uuS.}pdrto")S
c{4Ad(rS
, ?aLL( r:a.rL
AlmoP< foftbulrio"lS / AÀ»
QdCrÙtA,tL
(otvnZrz Ègl»S-
O
Ingredients:- Five10-ballblackchains- Five10-ballwhitechains
Trytoconnectthewhitebeadsandformthefive10-ballwhitechains…
Filltheschemewithfive10-ballblackAndfive10-ballwhitechains…
Rules:-NointersecDonallowed-Onebeadbelongstoonesinglechain
lNrdWAa,JS
I euertcy or rzi fi^l c
SPeq'es A A^ìt B
?AiLd i}t i "venx64'f o à S
A+{< 141744c7oJ e,ìafzb( oÈ
'----= or -rt{€ S(Ec;e5 AA \tr^t42zt,.,y.gJ o s- o,l rft-r ?aogeyt'drT
oF ÉdNJ I F t À" oa .È "4s
ila6l eao45
U+ 7 Uqn 4^t Uog 4e
Un* )\)so \ 3Uoe ) +
.?fu
U6 = t)eeh + rJeB 4,
{HiS iS 1+1€. tlr€n+qioiB€np€E{ flo Fro.òorzzrc5 ,
l^l ft W'ff (f -tllztrtz AL;nuLn ?tE Po53i Btz xÉfuPotsA T THz sA rtr dr1-? '
T+L€ *r/efW C< t'JreYl*a:9n)r'§ e U* rdìr*r ?getÀ0 tW C@fU\lAaoJNonbetL
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I euertcy or rzi fi^l c
SPeq'es A A^ìt B
?AiLd i}t i "venx64'f o à S
A+{< 141744c7oJ e,ìafzb( oÈ
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oF ÉdNJ I F t À" oa .È "4s
ila6l eao45
U+ 7 Uqn 4^t Uog 4e
Un* )\)so \ 3Uoe ) +
.?fu
U6 = t)eeh + rJeB 4,
{HiS iS 1+1€. tlr€n+qioiB€np€E{ flo Fro.òorzzrc5 ,
l^l ft W'ff (f -tllztrtz AL;nuLn ?tE Po53i Btz xÉfuPotsA T THz sA rtr dr1-? '
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i,r-l{ts
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C+tAà6€-
?- U,/Y1
Eò}ÉzdY oJ rti xir..) 6 is tttt
$+ (^-d) ('u^u-unft -'ru')
(? a*s - UnR - Uee)7L Kor
t-_lftttr i ^trettqciias?Atut€Iw-
44,'* =
AU-.ix
AS-ix, -rx
tuN* *.%(4*)rv(44-\
AI-,* : A U*,* - T- AS prit =
=Ksr[k%q^+ *%0 4^).' r4*(a-4^l
r?nfi,t Kry1 ,i'q*[ $r-rl_t ?",ì-1 -1'T'l
^#U' tiqi
[, * r,' u'J
'l i; \l \,' u'\r' J \l'
lNrdWAa,JS
I euertcy or rzi fi^l c
SPeq'es A A^ìt B
?AiLd i}t i "venx64'f o à S
A+{< 141744c7oJ e,ìafzb( oÈ
'----= or -rt{€ S(Ec;e5 AA \tr^t42zt,.,y.gJ o s- o,l rft-r ?aogeyt'drT
oF ÉdNJ I F t À" oa .È "4s
ila6l eao45
U+ 7 Uqn 4^t Uog 4e
Un* )\)so \ 3Uoe ) +
.?fu
U6 = t)eeh + rJeB 4,
{HiS iS 1+1€. tlr€n+qioiB€np€E{ flo Fro.òorzzrc5 ,
l^l ft W'ff (f -tllztrtz AL;nuLn ?tE Po53i Btz xÉfuPotsA T THz sA rtr dr1-? '
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i,r-l{ts
Au-n =
C+tAà6€-
?- U,/Y1
Eò}ÉzdY oJ rti xir..) 6 is tttt
$+ (^-d) ('u^u-unft -'ru')
(? a*s - UnR - Uee)7L Kor
t-_lftttr i ^trettqciias?Atut€Iw-
44,'* =
AU-.ix
AS-ix, -rx
tuN* *.%(4*)rv(44-\
AI-,* : A U*,* - T- AS prit =
=Ksr[k%q^+ *%0 4^).' r4*(a-4^l
r?nfi,t Kry1 ,i'q*[ $r-rl_t ?",ì-1 -1'T'l
^#U' tiqi
[, * r,' u'J
'l i; \l \,' u'\r' J \l'
tptticzz»
L}lft'cz.3D
4
6
Z = C@Ql'»'el*Tr oJ lAnoer\
Arf€rrloa r.;,Jerur ftaL noÀotrc7- i 5a,LUÌ= L U*
L<Ùs7 = L dgL
fttr TorAL @€rlc'.(
u =?60 u*2\
W Affr.n rr"trd6
ni Xrit,l& lS:
\+ (vts 0.) =
L*Luas 4r'4-{ dea 6:) \
fJ- Z uou +r (r-+t*us6(, +l)U(t''** 4r
T ('-^T +te To-r &L E»€rU'{ ÈetuQi
O= 9g uas lnYoA LUoa = TtL, Ò ut QsJL
(*^#u*ù
A z=coordination number
lNrdWAa,JS
I euertcy or rzi fi^l c
SPeq'es A A^ìt B
?AiLd i}t i "venx64'f o à S
A+{< 141744c7oJ e,ìafzb( oÈ
'----= or -rt{€ S(Ec;e5 AA \tr^t42zt,.,y.gJ o s- o,l rft-r ?aogeyt'drT
oF ÉdNJ I F t À" oa .È "4s
ila6l eao45
U+ 7 Uqn 4^t Uog 4e
Un* )\)so \ 3Uoe ) +
.?fu
U6 = t)eeh + rJeB 4,
{HiS iS 1+1€. tlr€n+qioiB€np€E{ flo Fro.òorzzrc5 ,
l^l ft W'ff (f -tllztrtz AL;nuLn ?tE Po53i Btz xÉfuPotsA T THz sA rtr dr1-? '
T+L€ *r/efW C< t'JreYl*a:9n)r'§ e U* rdìr*r ?getÀ0 tW C@fU\lAaoJNonbetL
i,r-l{ts
Au-n =
C+tAà6€-
?- U,/Y1
Eò}ÉzdY oJ rti xir..) 6 is tttt
$+ (^-d) ('u^u-unft -'ru')
(? a*s - UnR - Uee)7L Kor
t-_lftttr i ^trettqciias?Atut€Iw-
44,'* =
AU-.ix
AS-ix, -rx
tuN* *.%(4*)rv(44-\
AI-,* : A U*,* - T- AS prit =
=Ksr[k%q^+ *%0 4^).' r4*(a-4^l
r?nfi,t Kry1 ,i'q*[ $r-rl_t ?",ì-1 -1'T'l
^#U' tiqi
[, * r,' u'J
'l i; \l \,' u'\r' J \l'
tptticzz»
L}lft'cz.3D
4
6
Z = C@Ql'»'el*Tr oJ lAnoer\
Arf€rrloa r.;,Jerur ftaL noÀotrc7- i 5a,LUÌ= L U*
L<Ùs7 = L dgL
fttr TorAL @€rlc'.(
u =?60 u*2\
W Affr.n rr"trd6
ni Xrit,l& lS:
\+ (vts 0.) =
L*Luas 4r'4-{ dea 6:) \
fJ- Z uou +r (r-+t*us6(, +l)U(t''** 4r
T ('-^T +te To-r &L E»€rU'{ ÈetuQi
O= 9g uas lnYoA LUoa = TtL, Ò ut QsJL
(*^#u*ù
A z=coordination number
i,r-l{ts
Au-n =
C+tAà6€-
?- U,/Y1
Eò}ÉzdY oJ rti xir..) 6 is tttt
$+ (^-d) ('u^u-unft -'ru')
(? a*s - UnR - Uee)7L Kor
t-_lftttr i ^trettqciias?Atut€Iw-
44,'* =
AU-.ix
AS-ix, -rx
tuN* *.%(4*)rv(44-\
AI-,* : A U*,* - T- AS prit =
=Ksr[k%q^+ *%0 4^).' r4*(a-4^l
r?nfi,t Kry1 ,i'q*[ $r-rl_t ?",ì-1 -1'T'l
^#U' tiqi
[, * r,' u'J
'l i; \l \,' u'\r' J \l'
T' D^l+tJ
[ »NA G;»eA-se noJ
inr Aav€oÒS SaerJnoÀ aT "eG cottaPses
5e'.J S g Gt-a bu tE5 .
\l ur
-TUrtef
?a
t)Gnrr»alt-S :
. »rsA
'?eG, SalT
( u,Jo)
T? - D^lftl-It&o.l»^l*/rce/sor- - SAtr
Tt-ix + Tt *tr
=T rrtsrDÉ 1Ee D,,§*
Foor:;x fru ò^§&
€6gt)ir-ì Brr,.Cn co^ottoJSr*te C+tsnicAt- fsreanr A u
ìl$se A"§\ ql6rDt
C+ten, c+iL foi€rJ nl\L
godd\ 9r eaJlfi..)&fttg 6:9-terL 7a5S9ùnF
>N* cei< '
g0E
A,J§
T+IE
A DrEr..lt
A AF-.i^
'lr a^/
1lf,di+;
[,,O SnonC ?ÉggJ(w
water
1.9 M PEG 10 M PEG
water
1.9 M PEG 10 M PEG
T' D^l+tJ
[ »NA G;»eA-se noJ
inr Aav€oÒS SaerJnoÀ aT "eG cottaPses
5e'.J S g Gt-a bu tE5 .
\l ur
-TUrtef
?a
t)Gnrr»alt-S :
. »rsA
'?eG, SalT
( u,Jo)
T? - D^lftl-It&o.l»^l*/rce/sor- - SAtr
Tt-ix + Tt *tr
=T rrtsrDÉ 1Ee D,,§*
Foor:;x fru ò^§&
€6gt)ir-ì Brr,.Cn co^ottoJSr*te C+tsnicAt- fsreanr A u
ìl$se A"§\ ql6rDt
C+ten, c+iL foi€rJ nl\L
godd\ 9r eaJlfi..)&fttg 6:9-terL 7a5S9ùnF
>N* cei< '
g0E
A,J§
T+IE
A DrEr..lt
A AF-.i^
'lr a^/
1lf,di+;
[,,O SnonC ?ÉggJ(w
T' D^l+tJ
[ »NA G;»eA-se noJ
inr Aav€oÒS SaerJnoÀ aT "eG cottaPses
5e'.J S g Gt-a bu tE5 .
\l ur
-TUrtef
?a
t)Gnrr»alt-S :
. »rsA
'?eG, SalT
( u,Jo)
T? - D^lftl-It&o.l»^l*/rce/sor- - SAtr
Tt-ix + Tt *tr
=T rrtsrDÉ 1Ee D,,§*
Foor:;x fru ò^§&
€6gt)ir-ì Brr,.Cn co^ottoJSr*te C+tsnicAt- fsreanr A u
ìl$se A"§\ ql6rDt
C+ten, c+iL foi€rJ nl\L
godd\ 9r eaJlfi..)&fttg 6:9-terL 7a5S9ùnF
>N* cei< '
g0E
A,J§
T+IE
A DrEr..lt
A AF-.i^
'lr a^/
1lf,di+;
[,,O SnonC ?ÉggJ(w
DNAPEG
T' D^l+tJ
[ »NA G;»eA-se noJ
inr Aav€oÒS SaerJnoÀ aT "eG cottaPses
5e'.J S g Gt-a bu tE5 .
\l ur
-TUrtef
?a
t)Gnrr»alt-S :
. »rsA
'?eG, SalT
( u,Jo)
T? - D^lftl-It&o.l»^l*/rce/sor- - SAtr
Tt-ix + Tt *tr
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PEG3k PEG7k
PEG22k
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t
PHA sq Bgr^Ifry- ( Ui,to o^"), I ,uTIt€, &nb1;naÀS + AdD +" kn7 W@Lnr'JEa BY rilE
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intracellular phase-separation
WHAT HAVE WE LEARNED?
Ideal polymers behawe as a drunk (random valk!)
Styrofoam does not soak in coffee because
coffee is a poor solwent for polystyrene
(coil to globule)
Polymer mixtures can phase-separates,
forming droplets but also hydrogels
(contact lenses)