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Macromolecular CrowdingKeng-Hwee Chiam

Mathematical and Theoretical Biology Group

Goodsell (1994)Macromolecular Crowding, Oct. 15, 2003 – p.1/33

Outline

What: introduction, definition

Why: implications on cellular properties and processes

How: quantitative models

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Motivation

Biological media, e.g. cytoplasm:crowded with large moleculesbiophysical, biochemical, physiological propertiesaffected

in vitro assays:neglect non-ideal crowding by, e.g., using dilute solutionsresults cannot be extrapolated to in vivo situation

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Introduction

Cytoplasm packed with RNA, cytoskeletal elements, andother proteins that occupy 5%–40% of volume

D. discoideum, Medalia et al., Science (2002), E. coli, Goodsell (1999)

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Introduction

Red blood cells packed with hemoglobin, blood serumpacked with antibodies, albumin proteins, etc.

Goodsell (1999-2000)

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Definition

Sum of all macromolecule-macromolecule andmacromolecule-solvent interactions

Steric: repulsion due to excluded volumeHydrodynamic: viscous drag owing to motionChemical: hydrophilic and hydrophobic effectsvan der WaalsElectrostatic

Nonequilibrium: energy and number fluxes throughmembranes

Mesoscopic: finite-size effect

A messy situation!

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Steric repulsion

Presence of large molecules makes volume less accessible toother large molecules

Ellis (2001)

Energetically costly to add a large molecule to an alreadycrowded volume

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Steric repulsion

Neglect Coulombic interactions for now

Ideal fluid of point particles: PV = NkT

Fluid of hard spheres: P (V − b) = NkT

Excluded volume b is four times volume of spheres

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Steric repulsion

Presence of n hard spheres leaves (V − nv0) available for(n + 1)-th sphere

Number of microstates and entropy:

Ω ∝ V (V − v0) · · · [V − (N − 1)v0]

S = k log V + k log(V − v0) + · · · + k log[V − (N − 1)v0]

Equation of state:

P

T=

∂S

∂V=

Nk

V − Nv0/2, v0 =

4

3π(2a)3

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Implications

Transport in the cytoplasm

Kinetics of metabolic channeling

Efficiency of protein folding

Cellular volume regulation

Amyloid fibril formation

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Transport

Mechanisms for transport of viruses, gene carriers, etc. in thecytoplasm?

Diffusive or ballistic? Passive or active?

Important to:Rational design of gene deliveryBioengineering of biocompatible materialsInsights into intracellular reaction-diffusion-typemodeling

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Transport

Rate of self-diffusion of proteins lowered as crowdingincreases (Zimmerman and Minton, Annu. Rev. Biophys. Biomol. Struct. 22, 27 (1993)):

Replace D∇2c with ~∇ · [D(c)~∇c]

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Transport

How to obtain efficient transport in lieu of inefficientdiffusion?

Active propagation along cytoskeletal elements?

Crossover to active propagation as crowding increases

0 1

1

2

Volume fraction φ

Tra

nspo

rt e

xpon

ent γ

= d

log

⟨r2 (t)⟩

/ d

log

t

Replace ~∇ · [D(c)~∇c] with ~U(~x, c) · ~∇c + ~∇ · [D(c)~∇c]Macromolecular Crowding, Oct. 15, 2003 – p.13/33

Transport

Measure local transport properties of cytoplasm, e.g. Wirtz etal. (Biophys. J. 83, 3162 (2002), Biophys. J. 78, 1736 (2000)) and Sackmann etal. (Biophys. J. 76, 573 (1999), Biophys. J. 75, 2038 (1998))

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Transport

Inject tracers (e.g., fluorescent µspheres) into cytoplasm

Brownian motion with dissipation depending on earliervelocities:

mv(t) = −

∫

t

0

ζ(t − τ)v(τ)dτ + fR(t)

where ζ(t) is memory function

In thermal equilibrium, noise correlation:

〈fR(0)fR(t)〉 = kBTζ(t)

Obtain velocity correlation in terms of measurable meansquare displacement 〈∆r2(t)〉

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Transport

Susceptibility:

G(s) =kBT

πas〈∆r2(s)〉

Real part G′(ω) is storage modulus, imaginary part G′′(ω) isloss modulus

In a fluid, 〈∆r2(t)〉 = 6Dt with D = kBT/6πηa:

G′(ω) = 0, G′′(ω) = ηω

In a solid, 〈∆r2(t)〉 independent of t:

G′(ω) = const., G′′(ω) = 0

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Transport

Cytoplasm has non-zero G′(ω) and non-zero G′′(ω)

Stiff Hookean solid at high deformation rates

Soft viscous liquid at low deformation rates

Viscoelasticity arises from elastic macromolecules in viscousliquid

Replace diffusion in reaction-diffusion models with ???

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Metabolic channeling

How does crowding affect fluxes through signal transductionpathways?

Posphoenolpyruvate:carbohydrate phosphotransferase system(PTS):

Uptake and phosphorylation of carbohydratesPhosphoryl group transferred sequentially along a seriesof proteins to carbohydrate molecule

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Metabolic channeling

Rohwer, Westerhoff et al. (P. Natl. Acad. Sci. USA 95, 10547 (1998))

measured steady-state flux through PTS in vitro

Flux ∝ concentration2:low concentrationperfect channel, “hit and run”

Flux ∝ concentration:high concentration and/or with crowdingenzymes exist as complexes

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Metabolic channeling

Total flux and flux-response coefficient vs. total enzymeconcentration:

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Metabolic channeling

Two-enzyme group-transfer model:

Crowding added in as changes to association/disassociationrates

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Efficiency of protein folding

Proteins fold in central cage of GroEL/GroES chaperonin

Martin and Hartl (P. Natl. Acad. Sci. USA 94, 1107 (1997)) showed:crowding helps retain nonnative polypeptide

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Cellular volume regulation

How do cells detect changes in their volume?

Minton et al. (P. Natl. Acad. Sci. USA 89, 10504 (1992)) proposed thatduring swelling:

cellular interior less crowdedinhibits kinase relative to phosphatase activityincreases concentration of transporter

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Amyloid fibril formation

Neurodegenerative diseases, e.g., Parkinson’s disease,characterized by amyloid fibril formation

Shtilerman et al. (Biochemistry 41, 3855 (2002)) showed that crowdingreduced lag time for protofibril formation and conversion ofprotofibril to fibril

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Quantitative models

Hard spheres fluid, amenable to theory

Atomic-level models using molecular dynamics

Continuum models

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Hard spheres fluid

Toy model: Spheres with hard core potential

u(r) =

∞ r < a

0 r ≥ a

in vacuum

Two parameters: radius of spheres a, and volume fraction φ

Calculate thermodynamic and kinetic properties in (a, φ)phase space using:

tabulated virial coefficients (Hall and Minton, Biochim. Biophys. Acta

1649, 127 (2003))

Monte Carlo simulations

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Hard spheres fluid

Consider effects on equilibrium of dimerization

A + A A2

Calculate non-ideal contribution to K vs. volume fraction φ

(Hall and Minton, Biochim. Biophys. Acta 1649, 127 (2003))

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Hard spheres fluid

In addition to hard spheres, to study transport:

Mimic actin elements by stationary rods

Model binding and transport on these rods

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Atomic-level models

Elcock (P. Natl. Acad. Sci. USA 100, 2340 (2003)) modeled escape ofrhodanese from cavity of GroEL into crowded solvent

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Atomic-level models

Uses known GroEL conformation

Omits GroES

Models crowding agent as spheres

Incorporates repulsive r−12 interaction between spheres andrhodanese

Uses Brownian motion for crowder spheres

Assumes periodic boundaries

Contains 64000 atoms

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Atomic-level models

Crowding enhances efficiency of chaperonins GroEL andGroES:

Enhances association of GroEL and GroES, preventingescape

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Atomic-level models

Friedel et al. (J. Chem. Phys. 118, 8106 (2003)) simulated a version ofthe 46 residue off-lattice minimalist model using moleculardynamics

Restricts protein to sphere with soft well repulsive potential

Crowding decreases folding time

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Conclusions

Macromolecular crowding has important biophysical,biochemical, and physiological implications on intracellularprocesses

Interesting problems:Spatial dependence of transport in cytoplasm fromincreasing accurate modelingUnderstanding role of crowding in kinetics of metabolicchanneling

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