Lec 2

Question 1

Rank by persistence length
DNA
Vimentin
Actin
Microtubules
uncooked spaghetti

Answer 1

DNA~ 50 nm
Vimentin ~ 1µm
Actin ~10 µm
Microtubules ~ 5 mm
uncooked spaghetti ~10^18 m

Question 2

What is flexural rigidity?

Answer 2

A measure of a filament’s resistance to bending; higher means stiffer

Question 3

What does a larger 𝜅𝑓 mean in terms of filament shape?

Answer 3

The filament resists bending more and appears straighter

Question 4

What is persistence length 𝐿𝑝?

Answer 4

The distance over which filament tangent vectors remain correlated; a measure of stiffness

Question 5

What does it mean when 𝐿𝑝 ≫ 𝐿𝑐?

Answer 5

The filament behaves like a rigid rod

Question 6

What does it mean when 𝐿𝑝 ≪ 𝐿𝑐?

Answer 6

The filament is highly flexible and adopts coiled, entropic configurations

Question 7

What is contour length 𝐿𝑐?

Answer 7

The actual arc-length of the polymer (think of it as its full length when straightened)

Question 8

What’s the significance of the ratio 𝐿𝑝/𝐿𝑐?

Answer 8

It defines the mechanical regime: flexible (≪ 1), semi-flexible (~1), rigid (≫ 1)

Question 9

Why do flexible filaments appear coiled or crumpled?

Answer 9

They have many low-energy conformations, and entropy favors disordered shapes

Question 10

What happens when a force is applied to a flexible filament?

Answer 10

It loses entropy (fewer configurations), so a restoring force resists extension: entropic elasticity

Question 11

What is an entropic spring?

Answer 11

A flexible polymer whose restoring force comes from entropy, not just energy

Question 12

What determines the stiffness of an entropic spring?

Answer 12

It’s proportional to 𝑘𝐵𝑇𝐿𝑝/𝐿𝑐; longer or more flexible = softer spring

Question 13

What kind of energy scale does flexural rigidity operate in?

Answer 13

On the order of a few 𝑘𝐵𝑇 — meaning thermal fluctuations can significantly bend filaments

Question 14

What is buckling?

Answer 14

A mechanical instability where a compressed rod suddenly deforms laterally

Question 15

When does a filament buckle under compression?

Answer 15

When compressive force exceeds 𝐹buckling = (π²𝜅𝑓𝐿²)

Question 16

What happens to microtubules under compressive forces in cells?

Answer 16

They may buckle unless stabilized or bundled, due to their moderate stiffness and length

Question 17

Why do shorter filaments resist buckling better?

Answer 17

Because 𝐹buckling ∝ 1/𝐿² — shorter rods need more force to buckle

Question 18

What is the Worm-Like Chain (WLC) model?

Answer 18

A statistical mechanics model that describes bending elasticity of semi-flexible polymers like DNA and actin

Question 19

How does the end-to-end distance 𝑟𝑒𝑒 relate to 𝐿𝑝 and 𝐿𝑐?

Answer 19

For very flexible chains, ⟨𝑟𝑒𝑒²⟩¹/² ∝ 2𝐿𝑝𝐿𝑐; for stiff chains ≈ 𝐿𝑐

Question 20

What does it mean when tangent vector correlation drops to 1/e?

Answer 20

The separation is one persistence length—this defines 𝐿𝑝

Question 21

What is the behavior of ⟨𝑟𝑒𝑒⟩ for flexible polymers as they get longer?

Answer 21

It grows slower than length—more like a random walk

Question 22

What does a longer persistence length mean biologically?

Answer 22

The filament is straighter and less thermally wiggly

Question 23

Rank these by persistence length (shortest to longest): DNA, vimentin, actin, microtubules

Answer 23

DNA < vimentin < actin < microtubules

Question 24

What are the vibes of a flexible polymer?

Answer 24

Wiggly, entropically dominant, easily bent by thermal noise

Question 25

What are the vibes of a rigid rod-like filament?

Answer 25

Clean arcs, low entropy, strong mechanical integrity

Question 26

What is the Langevin function used for?

Answer 26

It models extension of semi-flexible chains under force (used in WLC model)

Question 27

What causes the 'springy' behavior of tangled biopolymers?

Answer 27

Reduction in entropy when stretched, leading to an effective entropic force

Question 28

Why does force-extension deviate from Hooke’s Law at large extensions?

Answer 28

Because the chain becomes taut and entropy is exhausted—force diverges as 𝑥 → 𝐿𝑐

Question 29

What does 'ideal scaling' mean for long flexible polymers?

Answer 29

⟨𝑟𝑒𝑒²⟩¹/² ∼ 𝐿𝑐¹/²: a square-root relationship typical of random walks

Question 30

Why is moment of inertia important in cytoskeletal filaments?

Answer 30

It determines how strongly a given shape resists bending — depends on cross-section geometry

Question 31

What is contour length (L_c)?

Answer 31

The full arc-length of the polymer — imagine straightening the entire filament along its backbone and measuring it end to end. Think of this as the 'true length' of the filament.

Question 32

What is end-to-end distance (r_ee)?

Answer 32

The straight-line vector between the two ends of the polymer. It’s the net displacement, not the full length. ## Footnote If the filament is wiggly, r_ee < L_c.

Question 33

What happens to entropy when stretched

Answer 33

decreases

Question 34

What makes buckling more likely

Answer 34

Longer, less flexible fiber buckles

Question 35

Force generation brownian ratchet actin vs mt vs intermediate

Answer 35

Actin filament polymerization(ATP) ~0.5-2.5 pN Microtubule polymerization (GTP) ~3-4 pN IF -> passive no force