trapping
Dielectric Optical particles in an inhomogeneous electric field are pulled in the direction of increasing field strength
To measure molecular forces optical traps are used which allow to manipulate very small particles under a microscope and to measure forces in the picoNewton range. Such forces can for example be measured during single molecular conformation changes occurring during muscle contraction.
Optical Trap
Biophysical Methods
Slide 1
The force of light on a diffracting particle
Optical Trap
Biophysical Methods
Slide 2
1
Optical Trap
Biophysical Methods
Slide 3
Optical trapping - ray optics and momentum
Optical Trap
Biophysical Methods
Slide 4
2
Optical trapping of a small particle can be understood using simple electrostatics
F-q E
+q s
F+
F
Assume an electric dipole in an electric eld. If the eld is homogeneous a torque will be exerted orienting the dipole but no net force will occur because the net forces on the positive and negative charges will cancel exactly.
r r F+ = qE
r r F! = !qE
However, if the eld is inhomogeneous then F+ and F- will not cancel and a net force occurs:
rr r r r r F = F+ + F! = q( E+ ! E! ) = q" E
Optical Trap
E is the difference between the eld at the plus and minus ends of the dipole Biophysical Methods Slide 5
Optical trapping of a small particle can be understood using simple electrostatics
rr r r r r F = F+ + F! = q( E+ ! E! ) = q" E
+q s -q E
F+
F-
For a small dipole the change of the electric eld between the position of q and +q is approximately linear and we can write E giving
E( x + ) = E (x ! ) +
"E "E (x ! x ! ) = E( x ! ) + s "x + "x x
E+ E-
"E dE = E ( x+ ) ! E ( x! ) = s "x x
For a eld in x direction which varies as a function of x, y and z we may then write:
x-
x+
x
dE x =
!E x !E !E r # !E !E !E & r r sx + x sy + x sz = s " % x , x , x ( = s " ) " E x !x !y !z $ !x !y !z '
If the electric eld has components also in y- and z-directions we get:
rr dE y = s ! " ! Ey
Optical Trap
and
rr dEz = s ! " ! Ez
or in vector form
r rrr dE = ( s ! " )E
Slide 6
Biophysical Methods
3
Optical trapping of a small particle can be understood using simple electrostatics
rr r r r r F = F+ + F! = q( E+ ! E! ) = q" E
The force can thus be expressed as:
+q s -q E
F+
r rrr dE = ( s ! " )E
F-
r r rrr F = q ! dE = ( p ! ") E
r r p =! "E
r rrr F = "( E # $) E
For dielectric material the dipole moment is induced by the applied eld: r is the polarizability of the material and is related to the index of refraction F= Giving for the force:
! rrr r rr rrr ! (E " E ) = 2 E # (! # E ) + 2( E " ! )E
Light is an electromagnetic wave So our force Trap r Optical is
rrv rrvvrr rrv vrr ! ( A " B) = A # (! # B) + B #( ! # A ) + ( A " !) B + (B " ! ) A
or
We now apply the product rule for vector derivatives:
rr E = E 0 ! cos"t
( E ! ") E = 1 "( E ! E ) # E $ (" $ E ) 2
and changes sign so only the quadratic term is not zero Slide 7
rrr
rrr
r
r
r
r r r ! r r r Biophysical Methods F = ! ( E " # )E = #( E " E) 2
+q s -q E
F+
F-
rr r r r r F = F+ + F! = q( E+ ! E! ) = q" E r r rrrr dE = ( s ! " )E p = ! " E
r !r r r ! r r2 F = " (E # E ) = "( E ) 2 2 r r2 1 rr2 2 = E0 (r ) ! cos "t = ! E0 ( r ) 2
In this formula we have to use the time average of E2:
r2 E
The light intensity is proportional to E2:
I=
1 ! cE 2 20 0
We then get the force:
r !rr F= #I(r ) 2" 0 c
r r p =! "E
r r P = N" # E
The atomic polarizability gives rise to a polarization
With an atomic density of N atoms/unit volume we get a total polarization:
How does the polarizability relate to the refractive index of the particle? Optical Trap Biophysical Methods
!
Slide 8
4
+q s -q E
F+
r r p =! "E
F-
r !rr F= #I(r ) 2" 0 c
With an atomic density of N atoms/unit volume we get a total polarization per unit volume of:
r r r P = Np = N" # E
But the eld E acting on a particular atom is not just the eld from the laser beam but also includes the eld from the surrounding atoms elsewhere in the trapped particle. We assume our trapped particles are spheres (e.g. beads) r The electric eld inside a homogeneously polarized sphere is Giving
r %r P ( r P = N" # ' E + * 3$0 ) &
! r P E Sphere = " 3#0
r r N#P r P" = N#E 3$0 ! r N# r P= E N# 1" 3$0 !
Optical Trap
r r with P = "0 # e E
"0 we get # e = N$ 1% 3"0
!
N$
solving for " & ) # e (1$ N" 3% + = N" % ' 0* 0 &N N ) "( + #e + = #e ' %0 3%0 * 3% # e "= 0 N #e + 3
Biophysical Methods
Slide 9
!
!
+q s -q E F+
r r p =! "E
F-
r "rr F= $I( r ) 2#0c
"=
3#0 $ e N $e + 3
!
Giving what is known as Clausius-Mosotti Formula
!
The electrical susceptibility is related to the dielectric constant
"r = 1+ # e
3# # $1 "= 0 r N #r + 2
!
Remember from the first spectroscopy lecture: !
n"
# = #r #0
Which gives us the relation
"=
3#0 n 2 $1 N n2 + 2
!
And we get the force
r 3"0 n 2 #1 1 r r 3 n 2 #1 1 r r ! F = N n 2 + 2 $ 2" c %I( r ) = 2N n 2 + 2 $ c %I( r ) 0
Biophysical Methods Slide 10
Optical Trap
!
5
+q s -q E
F+
F-
r 3 n 2 "1 1 r r F= # $I( r ) 2N n 2 + 2 c
!
The volume of a sphere with radius R is
VSphere =
43 "R 3
And the total force on the sphere becomes
!
r 2" $ n 2 #1 ' 3 r r F= & 2 ) * R +I( r ) c % n + 2(
!
This is for a sphere in vacuum. In our case we have a sphere in a surrounding medium and we must take the ratio of the refractive indices m=nS/nM. nS=refective index of sphere, nM=refractive index of surrounding medium. 2 R = radius of sphere 3 Typical values are nS=1.46 and nM=1.33 (water) giving m2~1.2
r 2! # m " 1 & r r % F= R (I (r ) c $ m 2 + 2'
Slide 11
Optical Trap
Biophysical Methods
How big is the force?
r 2! # m 2 " 1 & 3 r r % F= R (I (r ) 2 c $ m + 2'
For 1000 nm wavelength the diameter ds of the diffraction limited focal spot is about 1 m, the radius rs 0.5 m. To get an estimate we use the simple triangular intensity distribution
I = Ipeak (1 ! r / rs ),
The gradient is
for r < rs
r "I 1 "I "I !I = , , "r r "# "z
For displacement in focal plane:
The force thus is
I peak !I =" !r rs
Giving a force:
r 6P " m2 ! 1 % R 3 $ F= c # m2 + 2 & rs3
Typical values are nS=1.46 and nM=1.33 (water) giving m2~1.2
r 2! # m 2 " 1 & 3 I peak % F= R c $ m 2 + 2' rs
To relate this to the laser power we use the relation
I=
And integrate over the spot giving
dP da
r 6P 1 R 3 F= 3 c 0.06 rs
For a nm 100 diameter bead and 2 W laser power The force is about 2.5 pN
Slide 12
I peak
Optical Trap
3P =2 !rs
Biophysical Methods
6
Optical trapping
relative intensity
1.0 0.8 0.6 0.4 0.2 0.0 -10 -5 0 v (optical units) 5 10 h =(2J 1(v)/v)
2
2
But, the true intensity function is given by the point spread function giving a force that is position-dependent
r 2! # m 2 " 1 & 3 r r % F= R (I (r ) 2 c $ m + 2'
1.0 0.5
force pN
trapping force wavelength: laser power: objective NA: bead diameter:
1000 nm 1W 1.4 50 nm
In the central part the force is a linear function of displacement
Optical Trap
0.0 -0.5 -1.0 -1000nm -500 0 r (nm) 500 1000
Biophysical Methods
Slide 13
Optical trap measurement of single actin-myosin interactions
ATTACHED: At the start of the cycle shown in this gure, a myosin head lacking a bound nucleotide is locked tightly onto an actin lament in a rigor conguration (so named because it is responsible for rigor mortis, the rigidity of death). In an actively contracting muscle, this state is very short-lived, being rapidly terminated by the binding of a molecule of ATP. RELEASED. A molecule of ATP binds to the large cleft on the "back" of the head (that is, on the side furthest from the actin lament and immediately causes a slight change in the conformation of the domains that make up the actinbinding site. This reduces the afnity of the head for actin and allows it to move along the lament. (The space drawn here between the head and actin emphasizes this change, although in reality, the head probably remains very close to the actin.) COCKED: The cleft closes like a clam shell around the ATP molecule, triggering a large shape change that causes the head to be displaced along the lament about 5 nm. Hydrolysis of ATP occurs, but the ADP and inorganic phosphate (Pi produced remain tightly bound to the protein. FORCE-GENERATING: A weak binding of the myosin head to a new site on the actin lament causes release of the inorganic phosphate produced by ATP hydrolysis, concomitantly with the tight binding of the head to actin. This release triggers the power stroke, the force-generating change in shape during which the head regains its original conformation. In the course of he power stroke, the head loses its bound ADP, thereby returning to the start of a new cycle. ATTACHED: At the end of the cycle, the myosin head is again locked tightly to the actin lament in a rigor conguration. Note that the head has moved to a new position on the actin lament.
Optical Trap
Biophysical Methods
Slide 14
http://www.accessexcellence.org/RC/VL/GG/ecb/myosin_and_actin_model.html
7
Optical trap measurement of single actin-myosin interactions
Optical Trap
Biophysical Methods
Slide 15
Jeffrey T. Finer, Amit D. Mehta, and James A. Spudich (1995) Characterization of Single Actin-Myosin Interactions, Biophysical Journal 68: 291s-297s
Optical trap measurement of single actin-myosin interactions
Optical Trap
Biophysical Methods
Slide 16
Jeffrey T. Finer, Amit D. Mehta, and James A. Spudich (1995) Characterization of Single Actin-Myosin Interactions, Biophysical Journal 68: 291s-297s
8
Optical trap measurement of single actin-myosin interactions
Work during power stroke
Work during power stroke: 24 pN - nm (2.4 X x 10-20 J) 42 pN - nm (4.2 x 10-20 J). energy from ATP hydrolysis 10-19 J efciency 24-42%
Optical Trap
Biophysical Methods
Slide 17
Jeffrey T. Finer, Amit D. Mehta, and James A. Spudich (1995) Characterization of Single Actin-Myosin Interactions, Biophysical Journal 68: 291s-297s
Horse eosinophilic granulocytes have exceptionally large granules
About 60 granules per cell About 1.5 m diameter granules Release cytotoxic proteins to kill parasites
1m
Optical Trap Biophysical Methods Slide 18
9
For focal release of multivesicular contents Granule-granule fusion occurs inside the cell followed by compound exocytosis Fusion is preceded by tethering Model to measure tether Trap Optical forces
5.0 4.8 4.6 4.4 4.2
2.0 1.8 1.6 1.4 1.2 1.0
191.900 s
Biophysical1 9 2 . 0 1 9 2 . 2 Methods 191.8
192.4 time
192.6 (s)
192.8
193.0
Slide 19
193.2
Measuring tether forces
Optical Trap
Biophysical Methods
Slide 20
10
Isolating eosinophil granules
10 m
25 Passages
Eosinophil cells isolated from horse blood
Needle 25G11/2
(mM) 125 KCl 10 NaCl 7 MgCl 2 CaCl 5 EGTA 10 HEPES
Dnase I + Protease Inhibitors 1.5 ml Ependorf tube
Cells
2 m
Percoll Centrifugation
Debris Vesicles Intact cells
Optical Trap
Biophysical Methods
Slide 21
Optical Trap
Biophysical Methods
Slide 22
11
Trap stiffness calibration
Stage displacement
Stokes law
F= 6 . . r . v v F viscosity F= 6 . . r . v
:
coefficient of viscosity r : the particle radius v : the flow velocity
Objective
elastic spring
F=
.
x
F=
F spring
.x
: trap stiffness = 6 . . r . v
x
1.0 trapping force wavelength: laser power: objective NA: bead diameter: 1000 nm 1W 1.4 50 nm
x
force pN
Optical Trap
0.5 0.0 -0.5 -1.0
Biophysical Methods
-100nm
-50
0 r (nm)
50 100 Slide 23
Position detection for stiffness calibration
Displacement (!m)
200 160 120 80 40 0 5 10 15 20 25 time (s) 30 35
Displacement (!m)
200 160 120 80 40 6 8
Speed: 67 m/s
Speed:77 m/s
(a)
10 Time (sec) 12 14
(b)
(c)
Start movie 0.25 0.20 0.15 0.10 0.05 0.00 -0.05 -0.10 -0.15 -0.20 0 5 Displacement (m) Vertical position Horizontal position (a)
End movie 6 5 4 3 2 (b) (c) 1 50 55 0 0.00
10
15
20
25
30
35
40
45
Force (pN)
Optical Trap
Time (sec)
Biophysical Methods
Slide Displacement (m) 24
0.05
0.10
0.15
0.20
12
Single vesicle manipulation using an optical tweezers
moving trap fixed trap
25
V.Valero, T.Nevian, D.Ho and M.L., Biophysical Journal, in press.
Is the rate of separation force-dependent?
ksep
F=0 F>0
ksep
F=0 F>0
!E
!E (F) " !E (0) = "# $ F
log(k)
Force
Optical Trap
k sep(F) = k sep(0)$ e
Biophysical Methods
F$# k BT
Slide 26
13
Force step protocol for statistical analysis
displacement Trap displacement
pull push
0
Force vesicle-vesicle
20
Force (pN)
15 10 5 0 -5 50 Time (sec) 100
Optical Trap
Biophysical Methods
Slide 27
Movie and trace: 19Jan01, exper05
Optical tweezers data analysis
28
14
Tether dissociation consistent with disruption of a single critical bond
"E (F) # "E (0) = #x $ F ksep (F) = k sep (0) $ e x=
F$x kB T
kB T % k sep (F) ( ln' * = 2 # 3 F ' k sep (0) * & )
29
!
Tethering does not require cytosolic proteins
140
120 100 80 60 40 20 0 -1 0 1 2 3 4 force (pN) 5 6 7 8 9
The average work stretching the tether by 100 nm is ~4x10-19Nm or ~100 kT.
The average behavior of the tether is like that of an elastic spring with a spring constant of ~12 nm/pN.
Length (nm)
30
15