dinamica moleculara: introducere
DESCRIPTION
Dinamica moleculara: Introducere. In conditii fiziologice, biomoleculele sufera mai multe miscari si modificari Scara de timp a acestor miscari este intre femtosecunde si secunde Aceste miscari sunt esentiale pentru functia biomoleculelor. - PowerPoint PPT PresentationTRANSCRIPT
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Dinamica moleculara: Introducere
• In conditii fiziologice, biomoleculele sufera mai multe miscari si modificari
• Scara de timp a acestor miscari este intre femtosecunde si secunde
• Aceste miscari sunt esentiale pentru functia biomoleculelor
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Dinamica moleculara se bazeaza pe legea a doua a dinamici (legea lui
Newton):
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Miscarea atomilor x(t)
Se deduce din energia potentiala V(x)
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Energia potentiala V(x) a unei molecule include termeni de legatura
Deformarea legaturii
Indoirea unghiurilor
Rotatia de torsiune
Diedre improprii
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… si termeni de nelegatura
Interactiuni
electrostatice
Interactiuni
van der Waals
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Ecuatia pentru termenii de legatura din energia potentiala
)](cos1[)(
)()(
02
0
20
20
nAk
kllkRV
torsions
n
impropers
anglesbonds
lbonded
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Ecuatia pentru termenii de nelegatura
ijr
ji
ij
ij
ij
ij
jinonbonded r
r
r
r
rij
RV0
6min
12min
4])(2)[(()(
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• Fiecare dintre aceste interactiuni exercita o forta asupra unui atom dat din molecula
• Forta rezultanta se calculaza din functia energiei potentiale
Daca se cunoaste forta asupra unui atom se calculeaza miscarea:
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Dinamica moleculara: Introducere
La un moment de timp, t trebuie cunoscute:
• pozitia initiala a fiecarui atom
x1
• viteza
v1 = dx1/dt• si acceleratia
a1 = d2x1/dt2 = m-1F(x1)
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Pozitia x2 , a atomului dupa un interval de timp t este,
tvxx 112
tdx
dVmvtxFmvtavv x
1
111
11112 )(
iar viteza v2,
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Metoda diferentelor finite
...)(δt24
1)(δt
6
1)(δt
2
1 (t)vδt (t)rδt)(tr 432 tctbta
Se bazeaza pe dezvoltarea in serie Taylor.Pozitia este:
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Viteza (prima derivata):
...)(δt6
1)(δt
2
1 (t)aδt (t)vδt)(tv 32 tctb
Acceleratia (derivata a doua):
...)(δt2
1 (t)bδt (t)aδt)(ta 2 tc
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Algoritmul Verlet [Verlet 1967]
...)(δt2
1 (t)vδt (t)rδt)(tr
...)(δt2
1 (t)vδt (t)rδt)(tr
2
2
ta
ta
Prin adunarea celor doua ecuatii
)(δtδt)(tr- (t)r2δt)(tr 2 ta
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Vieza se poate calcula din
δt2
δt)(tr-δt)(tr(t)v
sau
δt
(t)r-δt)(trδt)
2
1(tv
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•Avantaj:
Necesar scazut de memorie: doua seturi de coordonate r(t) si r(t-δt) si acceleratia, a(t)
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•Dezavantaje:
Pozitiile r(t+δt) sunt obtinute din adugarea unui termen mic δt2a(t) la o diferenta de termeni mult mai mare
)(δtδt)(tr- (t)r2δt)(tr 2 ta
Vitezele sunt disponibile numai dupa calcularea pozitiilor din iteratia urmatoare
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•Dezavantaje:
Nu este un algoritm “self-starting”: poztiile sunt obtinute din pozitiile
actuale r(t) si pozitiile din iteratia precedenta r(t-δt)
)(δtδt)(tr- (t)r2δt)(tr 2 ta
La t=0 exista doar un set de pozitii si este necesara o ipoteza pentru pozitiile la t-δt
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•Se poate folosi seria Taylor
(0)vδt - (0)rδt)(r
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•Algoritmul Verlet al vitezelor [Swope et a. 1982]
δt)(ta(t)aδt 2
1 (t)vδt)(tv
(t)a δt2
1(t)vδt (t)rδt)(tr 2
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•Algoritmul “leap-frog” [Hockney 1970]
δt)2
1(tv δt)
2
1(tv
2
1(t)v
(t)aδt δt)2
1-(tvδt)
2
1(tv
δt)2
1(tvδt (t)rδt)(tr
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Configuratia initiala:
- Date experimentale(cristalografie de raze X, rezonanta magnetica nucleara)
- Modele teoretice
- Combinatie experiment-teorie
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Vitezele initiale, distributia Maxwell-Boltzmann
Tk
vm
Tk
mv
B
ixi
B
iix
21/2
2
1-exp
2)p(
2
2
2 2
)(-exp
2
1)p(
xx
x
Distrubutia Gauss (normala)
<x> este media si σ2 varianta: σ2 =<(x-<x>)2>
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Echilibrare
• Scop: aducerea sistemului din starea initiala la echilibru teremodinamic
• Se monitorizeaza diversi parametri (energia cinetica, potentiala, totala, viteze, temperatura, presiune)
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Productie
• Calcularea temperaturii:
cB
N
i i
NNTk
m
p
322
||E
1
2
cinetica
• Nc: numarul de constrangeri
• 3N-Nc: numarul gradelor de libertate
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Proprietati dependente de timp
• Functii de corelatie. Coeficienti de corelatie. M valori ale seturilor de date xi si yi
ii
M
iiixy yxyx
MC
1
1
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… sau normalizate (-1 ..+1)
i2
i2
ii
M
1ii
2M
1ii
2
M
1iii
yx
yx
yM
1x
M
1
yxM
1
xyc
Cxy= 0: nu exista corelatieCxy= 1: grad inalt de corelatie
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Intervalul de timp δt pentru integrarea ecuatiilor de miscare: 1 femtosecunda
Limitare datorata celei mai rapide forteScala de timp accesibila: 101-103 ns
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Ipoteza ergodica:
A A timpansamblu
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Cum se modifica o molecula in timpul MD
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In general, daca se cunosc valorile x1, v1 si energia potentiala V(x), traiectoria moleculara x(t) poate fi calculata din
tdx
xdVmvv
tvxx
ixii
iii
1
)(11
11
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MiscareaExtinderea
spatiala (nm)Tmpul
caracteristic (s)
Vibratia atomilor din legaturi chimice
0.2 ÷ 0.5 10-14 ÷ 10-13
Vibratii elastice ale domeniilor globulare
1 ÷ 2 10-12 ÷ 10-11
Rotatia laturilor laterale de la suprafata
0.5 ÷ 1 10-11 ÷ 10-10
“Libratia” de torsiunea a gruparilor din interior
0.5 ÷ 1 10-11 ÷ 10-9
Scala de timp a miscarilor in proteine (1)
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MiscareaExtinderea
spatiala (nm)
Tmpul caracteristic (s)
Miscarea realtiva a unor regiuni diferite (“hinge bending”)
1 ÷ 2 10-11 ÷ 10-7
Rotatia lanturilor laterale din interior
0.5 10-4 ÷ 1
Tranzitii alosterice 0.5 ÷ 4 10-5 ÷ 1
Denaturari locale 0.5 ÷ 1 10-5 ÷ 10
Plierea proteinelor ??? 10-5 ÷ 102
Scala de timp a miscarilor in proteine (2)
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Parametri esentiale pentru dinamica moleculara (setati de
utilizator)• Temperatura
• Presiune
• Pasul de integrare
• Constanta dielectrica
• Duratele echilibrarilor si ale productiei
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Dinamica la temperatura constanta
• Scop: studiul comportarii sistemului la schimbarile de temperatura (tranzitii de faza, “folding” si “unfolding”, “simulated annealing”)
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cB
N
i i
NNTk
m
p
322
||E
1
2
cinetica
Temperatura este legata de energia cinetica prin relatia:
Scalarea vietezelor [Woodcock, 1971]: daca temperatura la timpul t este T(t) si vitezele sunt multiplicate cu factorul λ, schimbarea de temperatura poate fi calculata:
)(/
)()1(
3
2
2
1)(
3
2
2
1
2
2
1
2
1
tTT
tTT
Nk
vm
Nk
vmT
new
B
iiN
iB
iiN
i
![Page 38: Dinamica moleculara: Introducere](https://reader035.vdocuments.mx/reader035/viewer/2022081417/56814e79550346895dbc138e/html5/thumbnails/38.jpg)
Model initial de ADN
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Model de AND cu ioni
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AND intr-o cutie de apa
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“Instantanee”
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“lungimea”
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Rasucirea medie
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Formarea unui helix
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Helix din 30 AA
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Plierea unei proteine
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Canal ionic
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Aquaporina
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Bistrat lipidic
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