Biomolecules as polymers

–at least two of the primary classes of biomolecules are polymers (proteins and DNA/RNA, often saccharides as well)

–physicists have developed a rich set of theories for polymers

–How can we use them to gain new biological understanding? DNA packing

into bacteria

Do atoms always matter?

PBP1a

For understanding the function of an enzyme, we need an atomic-detailed structural description

The bacterial cell wall (PBP’s substrate) is often best described from a statistical (average) point of view

Random walksA simple model of a polymer is a random walk in space

1D

3D

Fixed segments of equal length are connected at hinges, can go in any direction

length of each segment is defined as the Kuhn length (a)

PBoC 8.2 DNA as a random walkalso known as the Freely Jointed Chain (FJC) model

Measurable properties

What do we want to measure for these random walks?

“random” implies only average properties are useful, e.g., , , p(R,N)

Apply the tools of statistical mechanics to calculate overall length (macrostate) from its underlying microstates

counting # states (W) for each nr or nlPBoC 8.2.1

Probability distributions

nr0 N

P(nr;N)

P (nr;N) =N !

nr!(N � nr)!(1

2)N

What is P(R;N)?

p(R;N) =2

√

2πNe−R

2/2Na2

Binomial distribution

Gaussian distributionR-Na Na

P(R;N)

R = (nr - nl)*a end-end distance

p(R;N) =2

√

2πNe−R

2/2Na2

Normalization ( ) implies that

∫∞

−∞

p(R;N)dR = 1

P (R;N) =1

√

2πNa2e−R2/2Na2

and in 3D: P (R;N) = (3

2πNa2)3/2e−3R

2/2Na2

Distribution is Gaussian with mean = 0 and variance = Na2

PBoC 8.2.1

Probability distributions

Central limit theorem

For a set of N random variables {xn} with finite mean and variance (e.g., the individual segments of a polymer), the sum X = x1+x2+...xN (e.g., the end-to-end distance) will tend towards a Gaussian distribution regardless of the distribution of xn

P (R;N) =1

√

2πNa2e−R2/2Na2

sum of n dice throws

Persistence length (discrete)Persistence length (Lp) is defined microscopically by the correlation between the directions of successive segments of our chain (from Flory1:)

= Na2

Therefore Lp = a (the Kuhn length)

The random walk model we just developed is also known as the Freely Jointed Chain model

1Flory, Paul J. (1969). Statistical Mechanics of Chain Molecules. New York: Interscience Publishers

Lp = limN→∞

⟨N∑

j≥i

a⃗i ·a⃗j

a⟩

Lp =⟨R2⟩ + Na2

2Na

http://openlibrary.org/b/OL5613440M/Statistical_mechanics_of_chain_molecules

Persistence length (continuous)Persistence length (Lp) is defined microscopically by the correlation between the directions of successive segments of our chain:

Therefore Lp = a/2 (a is the Kuhn length)

This is the start of the worm-like chain model

h~t(s) · ~t(u)i = e�|s�u|/Lp

~R =

Z L

0ds ~t(s)

h~R2i = hZ L

0ds ~t(s) ·

Z L

0du ~t(u)i

= 2

Z L

0ds

Z L

sdu e�(u�s)/Lp ⇡ 2LLp = Na2 = aL

PBoC 8.2.1

= 2

Z L

0ds

Z L

sdu e�(u�s)/Lp ⇡ 2LLp = Na2 = aL

a = /Rmax in the continuous model

Radius of gyration

hR2Gi =1

N

NX

i=1

h(~Ri � ~RCM)2iRadius of gyration (RG): average distance between links and center-of-mass

~RCM =1

N

NX

i=1

~Riq

hR2Gi =r

LLp3

E. coli genome is ~ 4.6x106 base pairsq

hR2Gi ⇡ 5µm

About 2x as big as seen in the picture, but it has some constraints (e.g., circular, so it has to return to the cell body often)

Clearly strong forces are necessary to pack it in the cell!PBoC 8.2.2

Entropic force

Recall that our free energy G = U - TS + PV (Gibbs)

A change in energy (e.g., from applied tension) generates a restoring force, Frestore = -∂G/∂L

Even though the chain does not store internal energy, it still can exert a force as an entropic spring

PBoC 8.3.2

F = kTL/Na2F = -T(∂S/∂L)

U, T, P are constant for freely-jointed chain, making

Improvements to the FJC model

FJC model (fixed segments)

Worm-like chain (elastic, with an energetic bending cost)

modified WLC, now with a stretching cost as well

J. Hsin*, J. Strumpfer*, E. H. Lee, and K. Schulten. 


“Molecular origin of the hierarchical elasticity of titin: simulation, experiment and theory.” Annual Review of Biophysics, 40:187-203 (2011).

Worm-like chain (WLC) modelOur polymer is now characterized by a unit tangent vector t(s), where s is the position along the chain

How much energy does it cost to bend the chain?

PBoC 10.2.1

W (✏) =1

2E✏2 =

1

2E

✓�L

L0

◆2

AAACQ3icbZBPaxNBGMZnq/2XVk316OXFUEgvYXdbsJdCqQoeeqhgmmI2DbOTd5OhszPLzLuFsOS7efELePML9NKDUrwKTtIVte0DAw/P732ZmSctlHQUht+CpUePl1dW19YbG5tPnj5rbj0/daa0ArvCKGPPUu5QSY1dkqTwrLDI81RhL714M+e9S7ROGv2RpgUOcj7WMpOCk4+GzU+9doKFk8roHTiAKkqMH4d4Bu/+5OfxAs BfAonCjNpQJW9REYfjmh0PQ5glVo4ntHMeD5utsBMuBPdNVJsWq3UybH5NRkaUOWoSijvXj8KCBhW3JIXCWSMpHRZcXPAx9r3VPEc3qBYdzGDbJyPIjPVHEyzSfzcqnjs3zVM/mXOauLtsHj7E+iVl+4NK6qIk1OL2oqxUQAbmhcJIWhSkpt5wYaV/K4gJt1yQr73hS4jufvm+OY070W4n/rDXOjyq61hjL9kr1mYRe80O2Xt2wrpMsM/sin1nP4IvwXVwE/y8HV0K6p0X7D8Fv34DmDetwA==

E is Young’s modulus (units of force/area) ϵ is strain; W(ϵ) is strain energy density (energy/volume)

neutral axis

One way of visualizing the strain

Worm-like chain (WLC) modelPBoC 10.2.1

θ = s/(R+z) = L0/R

ΔL = s - L0 = (R+z)*L0/R - L0 = (z/R)*L0

W (✏) =1

2E

✓�L

L0

◆2=

1

2E⇣ zR

⌘2

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

Eb =

ZW (✏) dV = L0

ZE

2R2z2 dA

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

=EL02R2

Zz2 dA =

EIL02R2

AAACKXicbVDLSsNAFJ34rPVVdelmsAiuShIF3QhVERRcVLEPaNoymU7aoZNMmJkINcTPceOvuFFQ1K0/4qTNQlsPDBzOOZe597gho1KZ5qcxMzs3v7CYW8ovr6yurRc2NmuSRwKTKuaMi4aLJGE0IFVFFSONUBDku4zU3cFZ6tfviJCUB7dqGJKWj3oB9ShGSkudQvkYxudXHRM6XMdgbN+07SSBDg0UvG/bD7EjfNhNTmCag5dwKtopFM2SOQKcJlZGiiBDpVN4dbocRz4JFGZIyqZlhqoVI6EoZiTJO5EkIcID1CNNTQPkE9mKR5cmcFcrXehxoZ/ecKT+noiRL+XQd3XSR6ovJ71U/M9rRso7asU0CCNFAjz+yIsYVBymtcEuFQQrNtQEYUH1rhD3kUBY6XLzugRr8uRpUrNL1n7Jvj4olk+zOnJgG+yAPWCBQ1AGF6ACqgCDR/AM3sC78WS8GB/G1zg6Y2QzW+APjO8f1k6j8Q==

I is called the “geometric moment” (don’t have to calculate it!)

For a circle? Eb =EIL02R2

=EI(2⇡R)

2R2=

⇡EI

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

Persistence lengthThe WLC model represents a balance between internal energy (from the resistance to bending) and entropy

How does Lp vary with temperature?

PBoC 10.2.2

R θ

s Eb =EIL

2R2=

EIL

2

✓✓2

s2

◆

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

Let L = s for a short segment

=EI

2s✓2

AAACA3icbVDLSsNAFJ3UV62vqDvdDBbBVUmqoBuhKILuKtgHNLFMppN26OTBzI1QQsCNv+LGhSJu/Ql3/o3TNgttPTBwOOdc7tzjxYIrsKxvo7CwuLS8Ulwtra1vbG6Z2ztNFSWSsgaNRCTbHlFM8JA1gINg7VgyEniCtbzh5dhvPTCpeBTewShmbkD6Ifc5JaClrrl3jtOrG+xEOoTTqsoy7MCAAbmvds2yVbEmwPPEzkkZ5ah3zS+nF9EkYCFQQZTq2FYMbkokcCpYVnISxWJCh6TPOpqGJGDKTSc3ZPhQKz3sR1K/EPBE/T2RkkCpUeDpZEBgoGa9sfif10nAP3NTHsYJsJBOF/mJwBDhcSG4xyWjIEaaECq5/iumAyIJBV1bSZdgz548T5rVin1cqd6elGsXeR1FtI8O0BGy0SmqoWtURw1E0SN6Rq/ozXgyXox342MaLRj5zC76A+PzB04Jlq0=

s = ✓RAAAB83icbVBNS8NAEN34WetX1aOXxSJ4KkkV9CIUvXisYj+gCWWznbRLN5uwOxFK6d/w4kERr/4Zb/4bt20O2vpg4PHeDDPzwlQKg6777aysrq1vbBa2its7u3v7pYPDpkkyzaHBE5nodsgMSKGggQIltFMNLA4ltMLh7dRvPYE2IlGPOEohiFlfiUhwhlbyDb2mPg4AGX3olspuxZ2BLhMvJ2WSo94tffm9hGcxKOSSGdPx3BSDMdMouIRJ0c8MpIwPWR86lioWgwnGs5sn9NQqPRol2pZCOlN/T4xZbMwoDm1nzHBgFr2p+J/XyTC6CsZCpRmC4vNFUSYpJnQaAO0JDRzlyBLGtbC3Uj5gmnG0MRVtCN7iy8ukWa1455Xq/UW5dpPHUSDH5IScEY9ckhq5I3XSIJyk5Jm8kjcnc16cd+dj3rri5DNH5A+czx+qBpDK

Persistence length

Lp increases with increasing Young’s modulus (stiffer) Lp decreases with increasing temperature (more flexible)

PBoC 10.2.2

h~t(s) · ~t(u)i = e�|s�u|/Lp ! hcos(✓)i = e�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

Assume t(u) = t(0) = z axishcos(✓)i = e�s/Lp

AAACEnicbVA9SwNBEN2L3/ErammzGISkMN5FQRtBtLGwUDAxkIthbzNJluztHbtzQjjyG2z8KzYWitha2flv3HwUGn0w8Hhvhpl5QSyFQdf9cjIzs3PzC4tL2eWV1bX13MZm1USJ5lDhkYx0LWAGpFBQQYESarEGFgYSboPe+dC/vQdtRKRusB9DI2QdJdqCM7RSM1f0JVMdCdTnkSn42AVkRerrsXhC4S7dM/uXzXjQzOXdkjsC/Uu8CcmTCa6auU+/FfEkBIVcMmPqnhtjI2UaBZcwyPqJgZjxHutA3VLFQjCNdPTSgO5apUXbkbalkI7UnxMpC43ph4HtDBl2zbQ3FP/z6gm2jxupUHGCoPh4UTuRFCM6zIe2hAaOsm8J41rYWynvMs042hSzNgRv+uW/pFoueQel8vVh/vRsEsci2SY7pEA8ckROyQW5IhXCyQN5Ii/k1Xl0np03533cmnEmM1vkF5yPb3avnLM=

hcos(✓)i ⇡ h1� ✓2

2i = 1� 1

2h✓2i

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

small angle approximation

hEbi =EI

2sh✓2i

AAACJ3icbVDLSgMxFM34tr6qLt0Ei+CqzFRBN4ooBd1VsFXo1JJJb9vQTGZI7ghl6N+48VfcCCqiS//E9AXaeiBwOOdcbu4JYikMuu6XMzM7N7+wuLScWVldW9/Ibm5VTJRoDmUeyUjfBcyAFArKKFDCXayBhYGE26Bz0fdvH0AbEakb7MZQC1lLiabgDK1Uz576kqmWBFqsB9TXQ35C0+IV9SM7SNOC6fXoOOVjG5DdF8bRejbn5t0B6DTxRiRHRijVs69+I+JJCAq5ZMZUPTfGWso0Ci6hl/ETAzHjHdaCqqWKhWBq6eDOHt2zSoM2I22fQjpQf0+kLDSmGwY2GTJsm0mvL/7nVRNsHtdSoeIEQfHhomYiKUa0XxptCA0cZdcSxrWwf6W8zTTjaKvN2BK8yZOnSaWQ9w7yhevD3Nn5qI4lskN2yT7xyBE5I5ekRMqEk0fyTN7Iu/PkvDgfzucwOuOMZrbJHzjfP0/spQM=

= 2

✓kT

2

◆= kT

AAACDXicbVC7SgNBFJ31GeNr1dJmMAqxCburoE0gaGMZIS/ILmF2MpsMmX0wc1cIS37Axl+xsVDE1t7Ov3GSbKGJBwYO55zLnXv8RHAFlvVtrKyurW9sFraK2zu7e/vmwWFLxamkrEljEcuOTxQTPGJN4CBYJ5GMhL5gbX90O/XbD0wqHkcNGCfMC8kg4gGnBLTUM0+r2MGuYAGUcTZqYDfWaexMsCv5YAjnuIpHjZ5ZsirWDHiZ2DkpoRz1nvnl9mOahiwCKohSXdtKwMuIBE4FmxTdVLGE0BEZsK6mEQmZ8rLZNRN8ppU+DmKpXwR4pv6eyEio1Dj0dTIkMFSL3lT8z+umEFx7GY+SFFhE54uCVGCI8bQa3OeSURBjTQiVXP8V0yGRhIIusKhLsBdPXiYtp2JfVJz7y1LtJq+jgI7RCSojG12hGrpDddREFD2iZ/SK3own48V4Nz7m0RUjnzlCf2B8/gCIdJlV

2 DoF (one constrained by total length)

e�s/Lp ⇡ 1� s/LpAAACBXicbVC7TsMwFHXKq5RXgBEGiwqJpSUpSDBWsDAwFIk+pDZEjuu2Vh3Hsh1EFXVh4VdYGECIlX9g429w0wzQcqQrHZ1zr+69JxCMKu0431ZuYXFpeSW/Wlhb39jcsrd3GiqKJSZ1HLFItgKkCKOc1DXVjLSEJCgMGGkGw8uJ37wnUtGI3+qRIF6I+pz2KEbaSL69T+6Skjq+9sUYdpAQMnqALizBVPLtolN2UsB54makCDLUfPur041wHBKuMUNKtV1HaC9BUlPMyLjQiRURCA9Rn7QN5SgkykvSL8bw0Chd2IukKa5hqv6eSFCo1CgMTGeI9EDNehPxP68d6965l1AuYk04ni7qxQzqCE4igV0qCdZsZAjCkppbIR4gibA2wRVMCO7sy/OkUSm7J+XKzWmxepHFkQd74AAcARecgSq4AjVQBxg8gmfwCt6sJ+vFerc+pq05K5vZBX9gff4ACu+W/Q==

1� s/Lp = 1�1

2h✓2i = 1� 1

2

✓2skT

EI

◆

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

Lp =EI

kTAAAB/nicbVDLSgMxFM3UV62vUXHlJlgEV2WmCroRiiIouKjQF7TDkEkzbWgmGZKMUIYBf8WNC0Xc+h3u/BvTdhbaeiBwOOdc7s0JYkaVdpxvq7C0vLK6VlwvbWxube/Yu3stJRKJSRMLJmQnQIowyklTU81IJ5YERQEj7WB0PfHbj0QqKnhDj2PiRWjAaUgx0kby7YN7P4aXML25gz1hgjAdNbLMt8tOxZkCLhI3J2WQo+7bX72+wElEuMYMKdV1nVh7KZKaYkayUi9RJEZ4hAakayhHEVFeOj0/g8dG6cNQSPO4hlP190SKIqXGUWCSEdJDNe9NxP+8bqLDCy+lPE404Xi2KEwY1AJOuoB9KgnWbGwIwpKaWyEeIomwNo2VTAnu/JcXSatacU8r1Yezcu0qr6MIDsEROAEuOAc1cAvqoAkwSMEzeAVv1pP1Yr1bH7Nowcpn9sEfWJ8/r6yUrg==

Force-extension relationship for WLC

Ebend =EI

2

∫ L0

ds

R(s)2

partition function is summed over all possible curves (a path integral)

applying a force adds a term to the energy: Eapp. = −Fz = −F

∫ L0

tzds

⟨z⟩ =1

Z(F )

∫Dt(s)z[e−(Ebend+Eapp(F ))/kT ]

= kTd lnZ(F )

dFPBoC 10.2.3

=kTLp

2

∫ L0

|dt

ds|ds2

Z =

∫Dt(s)e−Ebend/kT =

∫Dt(s) exp(−

Lp

2

∫ L0

|dt

ds|ds)2

Good luck calculating this!

don’t assume R is constant

Force-extension relationshipA closed form solution for does not exist!

Marko, J. F. and Siggia, E. D. (1995). Stretching DNA. Macromolecules, 28, 8759–8770.

FLpkT

≈

z

L+

1

4(1 − z/L)2−

1

4interpolation formula

PBoC 10.2.3

deviates by up to 10% from numerical solution (not shown)

Force spectroscopy

DNA RNA protein

• different force profiles are like molecular signatures

PBoC 8.3.1

• deviations from WLC imply a change in structure

Single-molecule techniques

AFM optical tweezers

magnetic tweezers

pipette-based

PBoC 8.3

Atomic-force microscopy

Elias M Puchner, Hermann E Gaub, (2009) Force and function: probing proteins with AFM-based force spectroscopy, Current Opinion in Structural Biology, 19: 605-614.

unfolding of titin kinase + Ig domains

fits from WLC model

20-ns SMD Simulation of fibrinogen, 1.06 million atoms

A Blood Clot Red blood cells within a network of fibrin, composed of polymerized fibrinogen molecules.

B. Lim, E. Lee, M. Sotomayor, and K. Schulten. Molecular basis of fibrin clot elasticity. Structure, 16:449-459, 2008.

Atomic-force microscopy

Rico, Gonzalez, Casuso, Puig-Vidal, Scheuring. High-Speed Force Spectroscopy Unfolds Titin at the Velocity of Molecular Dynamics Simulations. Science, 342:741, 2013.

High-speed AFM show agreement with (relatively) slow simulations

Atomic-force microscopy

.01 Å/ns

optical tweezers

gradient induces lateral force

gradient induces axial force

unfocused laser focused laser

PBoC 4.3.1