Multiscale computational studies of biological light capture

(1)

(2)

+

(3)

(4)

(5)

(6)

+

(7)

(8)

(9)

(10)

+

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

Ψ = E Ψ,

Ψ E

= − !

i

1 2 ∇

²i

− !

A

1 2m

A

∇

²A

− !

i,A

Z

A

|

^A

−

ⁱ

| + !

i<j

1 |

ⁱ

−

^j

| + !

A<B

1 |

^A

−

^B

|

i, j A, B

m

Z Ψ

m e !

1

4πϵ0 m =e=!= _4πϵ¹₀ = 1

(20)

Ψ

= − !

i

1 2 ∇

²i

− !

i,A

Z

A

|

^A

−

ⁱ

| + !

i<j

1 |

ⁱ

−

^j

| ,

E = ⟨ Ψ | | Ψ ⟩

⟨ Ψ | Ψ ⟩ ≥ ⟨ Ψ

0

| | Ψ

0

⟩

⟨ Ψ

0

| Ψ

0

⟩ ,

Ψ

0

(21)

Ψ = φ

1

(1)φ

2

(2)...φ

N

(N),

Ψ φ

⟨ α | α ⟩ = ⟨ β | β ⟩ = 1

⟨ α | β ⟩ = ⟨ β | α ⟩ = 0,

α β

Φ = 1

√ N !

"

φ

1

(1) φ

2

(1) ... φ

N

(1) φ

1

(2) φ

2

(2) ... φ

N

(2) φ

1

(N ) φ

2

(N ) ... φ

N

(N )

"

,

(22)

i

φ

i

= ϵ

i

φ

i

,

ϵ

i

φ

i

= #

i

+ !

j

(

_j

−

^j

) $ φ

i

,

i

φ

i

= %

− 1

2 ∇

²i

− !

A

Z

A

|

^A

−

ⁱ

|

&

φ

i

j

φ

i

= ⟨ φ

j

| r

⁻_ij¹

| φ

j

⟩ φ

i

j

φ

i

= ⟨ φ

j

| r

_ij⁻¹

| φ

i

⟩ φ

j

N − 1

χ φ

i

= !

α

c

αi

χ

α

,

(23)

α

χ

ζ,n,l,m

(r, θ, φ) = N Y

l,m

(θ, φ)r

^2n−2−l ^−ζr²

,

N Y

l,m

(θ, φ) r

n l m ζ

χ

ζ,n,l,m

(r, θ, φ) = N Y

l,m

(θ, φ)r

ⁿ⁻¹ ⁻^ζr

,

(24)

−1

E = E

0

− E

(25)

E = !

i<j

!

a<b

( ⟨ φ

i

φ

j

| φ

a

φ

b

⟩ − ⟨ φ

i

φ

j

| φ

b

φ

a

⟩ )

²

ϵ

i

+ ϵ

j

− ϵ

a

− ϵ

b

, i, j a, b

M

⁵

M

(26)

Ψ = a

0

Φ + !

a Φ + !

a Φ + ... = !

i=0

a

i

Φ

i

, Φ Φ Φ

M

¹⁰

M

⁶

Ψ = Φ ,

=

1

+

2

+

3

+ ...

N

.

=

1

+

2

= 1 +

1

+ (

2

+

¹₂ ²₁

) + (

2 1

+

¹₆ ³₁

)...

(27)

2 2

1 2 1 3

1

(28)

N N

ρ

E = E [ρ].

E[ρ] = T [ρ] + E [ρ] + J[ρ] + E [ρ],

T [ρ] E [ρ]

J [ρ]

E [ρ]

E [ρ] = (T [ρ] − T [ρ]) + (E [ρ] − J[ρ]),

T [ρ] E [ρ]

(29)

E [ρ]

N

M

⁴

(30)

E

₋

= E + E . E

E

⁽²⁾

E

⁽²⁾

= !

AB

!

n=6,8,10,...

s

n

C

_n^AB

r

_ABⁿ

f

d,n

(r

AB

),

AB C

_n^AB

n

r

AB

s

n

f

d,n

M

³

M

⁴

ω

' ( ' (

= ω

' 1 0 0 − 1

( ' (

,

(31)

1/

1 r

12

= 1 − (µr

12

) r

12

+ (µr

12

) r

12

, µ

M

⁵

M

⁴

Y

_ai^Q

= c !

bj

t

^ab_ij

B

^Q_bj

,

(32)

c = 1.3 B

_bj^Q

t

^ab_ij

t

^ab_ij

= − )

P

B

_ai^P

B

_bj^P

ϵ

iajb

,

ϵ

iajb

i, j a, b

1 ϵ

iajb

=

*

∞ 0

( − ϵ

iajb

t) t ≈

nL

!

z

w

z

( − ϵ

iajb

t

z

) =

nL

!

z

w

z

( − ϵ

ia

t

z

) ( − ϵ

jb

t

z

),

w

z

t

z

ϵ

ia

= ϵ

i

− ϵ

a

ϵ

jb

= ϵ

j

− ϵ

b

t

^ab_ij

t

^ab_ij

= −

nL

!

z

w

z

!

P

B

^P_ai

( − ϵ

ai

t

z

) B

_bj^P

( − ϵ

bj

t

z

)

Y

_ai^Q

= − c

os nL

!

z

w

z

!

P

N

_z^{P Q}

B

_ai^P

( − ϵ

ai

t

z

)

N

_z^{P Q}

= !

bj

B

_bj^Q

B

_bj^P

( − ϵ

bj

t

z

),

n

L

n

L

(33)

β

i

= m

i i

= m

i 2 i

t

²

= −∇ U ( ),

(34)

i

i m

i i i

U

U = U + U ,

U = !

b

K

b

(b − b

0

)

²

+ !

θ

K

θ

(θ − θ

0

)

²

+ !

χ

K

χ

[1 + (nχ − σ)] + !

φ

K

θ

(φ − φ

0

)

²

U = !

i<j

% ϵ

ij

#% R

,ij

r

ij

&

12

− % R

,ij

r

ij

&

6

$ + q

i

q

j

r

ij

&

.

U b

θ χ φ

U i, j

q

ϵ

ij

R

,ij

U

(35)

E = E ( + ) + E ( ) − E ( ),

E = E ( ) + E ( ) + E

/

( + ),

E

/

(36)

ρ ρ = ρ + ρ ,

ρ ρ

ρ

(37)

(38)

Energy

isolated chromophore

strained chromophore

chromophore

in protein monomer dimer monomer

∆Estrain ∆Eel

GS ES

B

Energy

GS ES

C

Molecular geometry GS

ES

Energy

A

vertical (VEE)

adiabatic (0-0)

∆E ∆E

(39)

(40)

β

(41)

(42)

(43)

β α

−

→

(44)

GFP-A GFP-B

T62 E222 S205

H148

R96 V150

Q69

Q94 T203

−

A B C

P680 QB

QA

PhD2

PhD1

PD2

PD1

ChlD1 ChlD2

TyrZ

Mn4O5Ca Fe

P700

QA

A0A

PB

PA

FeX

Q1B

A0B

AA AB

P870 Fe

QB

BPhL BPhM

BChlL BChlM

PL PM

Q1A

(45)

+/•

(46)

PSB Lys polyene

β-ionone

A B

β

π

(47)

λ

π

+

(48)

KR2

₅₂₆

K

M

₄₀₀

D L

₅₀₅

O

₅₆₆

hν

Na

⁺

in Na

⁺

out

Na⁺

A B

E11 R109

D251 D116

N112 L120 S64

Q123 S60 N61

IN

OUT

PSB

+

− +

+

(49)

+

(50)

(51)

−

β

α α

β

ϵ

+/•

(52)

+/•

ϵ

β

β α

γ β

T

(53)

+

µ µ

+

+ +

T

+

ϵ ϵ

∼

⁻¹

(54)

(55)

(56)

-0.1 0 0.1 0.2 0.3 0.4

20 30

40 50

60

RVS-error (eV)

RVS-energy threshold (eV) p-HBDI

p-HBDI^- GFP-B

-0.1 0 0.1 0.2 0.3 0.4

20 30 40 50 60 100 150

200 anticore core full

RVS-error (eV)

RVS-energy threshold (eV) p-HBDI

p-HBDI^-

RVS error (eV)

RVS threshold (eV) RVS threshold (eV)

A B

−1

/

(57)

−1

−1 −1

−1

(58)

P680

P700

P870

2.03 eV 2.04 eV

2.00 eV 2.01 eV 2.02 eV 2.04 eV

2.03 eV

1.61 eV

2.02 eV 2.04 eV 2.05 eV 2.07 eV

1.52 eV 1.56 eV 1.59 eV 1.62 eV 2.03 eV

2.08 eV

1.52 eV 1.61 eV

BChl-L Chl-L Chl-L

(Chl-L)

2

(Chl-L)

4

(Chl-L)

2

(Chl-L)

4

(BChl-L)

2

(BChl-L)

4

(59)

ϵ

± e

β

(60)

(E /E ) (E /E )

E E

β β

β

+

µ

(61)

5 6 7 8 9 10

0 0.2 0.4 0.6 0.8 1 1.2 1.4

d(SB − L120) (Å)

Time (µs)

KR2 K/L M

A

B C D

E F

G

B

C D

E F

K/L → M

G

B

D

PSB L120 D116

N112

PSB L120

D116

N112

SB

L120 D116

N112

KR2 K/L M

A

S64

S60 N61

L120 Q123

E11 D116 N112 D251

R109

C

Open Closed

trans-PSB cis-PSB cis-SB

E11 R109

D251 N112

D116 L120

Q123 S64 S60

E11 E11

R109

N112 N112

D251 D116

L120

Q123 Q123

D116 L120

S64 S64

S60 S60

Z = 0 D251

−20

−15

−10

−5 0 5 10 15 20

0 0.1 0.2 0.3 0.4 0.5

Z (Å)

Time (µs) K/L

A

M

B

Site 1

Site 1 Site 2 Site 3

Site 2 Site 3

PSB PSB

SB IN

OUT

A B

µ

+

µ

+

(62)

Absorption (nm)

Intensity

350 400 450 500 550 600 650

0.000.010.020.03

KR2 L M O

λcalc = 500 λexp = 505

λcalc = 553 λexp = 566 λ_calc = 526

λ_exp = 526 λcalc = 417

λexp = 400

+

(63)

+

+ +

+

+ + +

(64)

(65)

+

(66)

(67)

(68)

(69)

(70)

ππ

(71)

(72)

←

(73)

(74)

(75)

β

(76)

+

+ −

(77)

(78)

(79)

(80)

→ →

(81)

(82)

(83)

+ +

(84)

+ + +

+

(85)

(86)

Y

(87)

(88)

(89)