Modeling Diffusion and Motion in Cells at the Molecular Level – Chapter 1. The Dynamics of α-Synuclein – Methods for α-Synuclein

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Modeling Diffusion and Motion in Cells at the Molecular Level – Chapter 1. The Dynamics of α-Synuclein – Methods for α-Synuclein

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Modeling Diffusion and Motion in Cells at the Molecular Level

Wendell Smith

Outline

  • The Dynamics of α-Synuclein and Disordered Proteins

    • Can we model a disordered protein without biasing it towards folding?

    • Compare model with smFRET experiments

    • Predict dynamics from model

  • Dynamics near the Glass Transition

    • What are the significant factors in dynamics in the bacterial cytoplasm?

    • Nucleoid effects, activity, crowding

Chapter 1. The Dynamics of α-Synuclein

Can we model a disordered protein without biasing it towards folding?

α-Synuclein

  • Function

    • Related to cellular signaling and control

    • Aggregation of α-synuclein linked to Parkinson’s disease and Lewy body dementia

  • Structure

    • Intrinsically disordered protein (IDP)

      • Does not have a well-defined static structure

    • More compact than a random coil of the same length

Previous Work

\[R_g = \sqrt{\frac{1}{N_p}\sum_i^{N_p} \left(\vec{r}_i - \left<\vec{r}\right> \right)^2}\]

  • Experimental methods for disordered proteins are limited

Previous Work

CHARMM Force Field at 523 K with NMR constraints!

All-Atom Models are calibrated for folded proteins, and are biased toward folding.

Can we simulate an IDP with a simple modeland arrive at realistic results?

Previous Work

  • AMBER and CHARMM

    • Standard protein force fields calibrated with crystal structures

    • Tends to bias towards folding [1]

  • ABSINTH, MARTINI

    • General and coarse-grained

  • Constrained models

    • Based on specific experimental data

Methods for α-Synuclein

Molecular Dynamics Simulations

Start with particles at initial positions \(\vec{r}_i\) and velocities \(\vec{v}_i\)

Calculate the forces on each particle, \(\vec{f}_i\)

  • This is where the model comes in!

Integrate numerically:

\[\begin{align*} \vec{r}_{i}(t + \delta t) & =\vec{r}_{i}(t) + \delta t\,\vec{v}_{i}(t) \\ \vec{v}_{i}(t + \delta t) & =\vec{v}_{i}(t) + \delta t\,\frac{1}{m_{i}}\vec{f}_{i}(t) \end{align*}\]

Repeat Steps 2–3 for 3×10¹¹ times or so, and this follows Newton’s equations:

\[\begin{align*} \frac{d \vec{r}_{i}}{dt} & = \vec{v}_{i} \\ \frac{d \vec{v}_{i}}{dt} & = \frac{1}{m_{i}}\vec{f}_{i}(t) \end{align*}\]
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Molecular Dynamics Simulations

  • Velocity Verlet for time reversibility and better energy conservation

  • We integrated the Langevin equation, to simulate an implicit solvent:

    \[\frac{d\vec{v}_{i}}{dt}=-\frac{1}{m_{i}}\vec{\nabla}_i U-\gamma \vec{v}_{i}+\sqrt{\frac{2\gamma k_{B}T}{m_i}}\Gamma\left(t\right)\]

    • \(-\gamma \vec{v}_{i}\) is a drag term

    • \(\Gamma\left(t\right)\) provides a random force

Building a Model

All-Atom

United-Atom

Coarse-Grained

Building a Model: Geometry

All-Atom and United-Atom

Coarse-Grained

Potential

Parameters

Potential

Parameters

Bond Lengths and Angles

Stiff Spring

PDB Data

Soft Spring

AA and UA probabilities

Dihedral Angles

ω only

ω = π

\(\sum a_{n}\cos^{n}\phi\)

AA and UA probabilities

Atom / Bead Sizes

Lennard-Jones Repulsive (WCA)

Refs. [2] and [3]

Lennard-Jones Repulsive (WCA)

\(\sigma=4.8\,Å\), from \(R_{g}\) of residues

Building a Model: Long-Range Interactions

Electrostatics Hydrophobicity
\[V_{ij}^{\textrm{es}}=\frac{1}{4\pi\epsilon_{0}\epsilon}\frac{q_{i}q_{j}}{r_{ij}}e^{-\frac{r_{ij}}{\ell}}\]

  • Coulomb interaction

  • Debye screening

  • Uses partial charges

\[V_{ij}^{a} \propto\left(\frac{\sigma^{a}}{R_{ij}}\right)^{12}-\left(\frac{\sigma^{a}}{R_{ij}}\right)^{6} R_{ij}>2^{\frac{1}{6}}\sigma^{a}\]

  • Attractive Lennard-Jones potential between \(\mathsf{C_{\alpha}}\) atoms

  • Relative hydrophobicities from tables

  • Overall energy scale unknown

  • Define \(\alpha\equiv\frac{\textsf{Hydrophobicity Energy}}{\textsf{Electrostatics Energy}}\)

    • a unitless free parameter

Full Model

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Results for α-Synuclein

All-Atom Geometry

All-Atom

PDB Structures

Zhou et al. [2] provided atom sizes calibrated to a hard sphere model

United-Atom Geometry

United-Atom

PDB Structures

Richards et al. [3] provided atom sizes calibrated to calculate packing densities; we multiplied by 0.9

Radius of Gyration (\(R_{g}\))

  • Black Solid: All-Atom

  • Red Dashed: United-Atom

  • Green Dotted: Coarse-Grained

  • Grey Area: Experimental Results

    • Average \(\left<R_g\right> \approx 33\,\textrm{Å}\)

\[\alpha=\frac{\textrm{Hydrophobicity Strength}}{\textrm{Electrostatic Strength}}\]

smFRET

Single-Molecule Förster Resonance Energy Transfer

smFRET of α-synuclein

smFRET Comparison (United-Atom)

  • Black: Experiment

  • Red: Geometry (Random Walk)

  • Green: Globule (\(\alpha \gg 1\))

  • Blue: Electrostatics (\(\alpha = 0\))

  • Purple: Our Model (\(\alpha = 1.1\))

\[F_{\textrm{eff}}=\left\langle \frac{1}{1+\left(\frac{R_{ij}}{R_{0}}\right)^{6}}\right\rangle\]

smFRET Comparison (Coarse-Grained)

  • Black: Experiment

  • Red: Geometry (Random Walk)

  • Green: Globule (\(\alpha \gg 1\))

  • Blue: Electrostatics (\(\alpha = 0\))

  • Purple: Our Model (\(\alpha = 1.1\))

\[F_{\textrm{eff}}=\left\langle \frac{1}{1+\left(\frac{R_{ij}}{R_{0}}\right)^{6}}\right\rangle\]

smFRET Comparison

United-Atom

Coarse-Grained

  • Black: Experiment

  • Purple: Our Model

  • Red: Geometry

  • Blue: Electrostatics

  • Green: Globule

Comparison to Constrained Simulations

◼ Red Squares: Our simulation

▲ Blue Triangles: Constrained simulation

◼ Closed: Constrained pairs

◻ Open: Unconstrained pairs

Average distance between pair i–j
Standard deviation between pair i–j

Conclusion

  • We can use a simple, 2-term model to study the conformational dynamics of α-synuclein calibrated to experiments

  • This model accurately predicts experimental results

  • The structure of α-synuclein is intermediate between a random walk and a collapsed globule

Chapter 2. Disordered Proteins

Can we extend this model to other disordered proteins, and use it to understand their dynamics?

Disordered Proteins

Charge vs. Hydrophobicity

● Green Circles: Known IDPs

◻ Purple Squares: Folded Proteins

Absolute value of the electric charge per residue Q versus the hydrophobicity per residue H

  • Uversky et al. [4] showed that charge and hydrophobicity were predictors of disordered proteins

  • They drew a line at \(Q=2.785H-1.151\)

smFRET Comparisons

  • Black: Experiment

  • Red: Our Model

  • Purple: Just Hydrophobicity

  • Blue: Just Electrostatics

Radius of Gyration (\(R_g\))

  • Black: Experiment

  • Green: Our Model

  • Blue: Electrostatics

  • Purple: Hydrophobicity

Radius of Gyration (\(R_g\)) Scaling

\[R_g(N_p) = \sqrt{\frac{1}{N_p}\sum_i^{N_p} \left(\vec{r}_i - \left<\vec{r}\right> \right)^2}\]

\(R_g(n)\) is calculated over portions of the protien of length n and averaged over time

Radius of gyration of 5 proteins

Scaling of partial \(R_g\) with chemical distance

Radius of Gyration Scaling

Scaling exponent ν with distance d from charge-hydrophobicity line

Scaling of partial \(R_g\) with chemical distance

Conclusion

  • This model can extend to other disordered proteins

  • Hydrophobicity plays a very strong role in IDP dynamics, with electrostatics relevant to some proteins

  • We can use the average hydrophobicity and charge of residues to predict the overall dynamics of IDPs

Chapter 3. Dynamics near the Glass Transition

What are the significant factors in dynamics in the bacterial cytoplasm?

Dynamics in Cells

  • Cells are full of large molecules, which may have an effect on particle dynamics

  • These macromolecules may take up anywhere from 5% to 40% of volume

    • Including bound water, these estimates could go as high as 50% to 60%, well into the glass transition region for hard spheres

  • Sub-diffusive and non-Gaussian behavior has been observed in particle motions in the cytoplasm

Dynamics in Cells

Diffusion of a large, fluorescent protein (GFP-μNS) in the cytoplasm of Escherichea Coli

Wild-type

Inactive metabolism

GFP-μNS is the avian reovirus protein μNS attached to Green Fluorescent Protein

Colors represent particle size. Figure from Parry et al. [5]

  • None of these tracks is diffusive (slope 1)

  • Small particles behave differently than large particles

  • Metabolic activity has a significant effect on particle dynamics

Nucleoid Effects

  • Bacterial DNA aggregates in the "nucleoid" region

  • How does this affect dynamics?

Nucleoid Effects

Locations of GFP-μNS particles

60 nm diameter
95 nm diameter
150 nm diameter

Data from Ivan Tsurovtsev, Jacobs-Wagner laboratory

Dark: more GFP-μNS

Light: less GFP-μNS

GFP-μNS particles are excluded from the nucleoid region

Models

Hard Nucleoid Soft Nucleoid

Model the nucleoid as an excluded volume region, which particles can go around

Derive a potential along the x-axis to "push" particles out of the nucleoid

Hard Nucleoid Results

  • Behavior is highly dependent on nucleoid size and particle size

    • Large particles cannot travel from pole to pole

    • Medium particles display intermediate behavior

    • Small particles diffuse freely

The hard nucleoid was modeled with a length of 2 μm and a radius of 0.7 μm (thin lines), 0.75 μm (medium lines), and 0.8 μm (thick lines).

Soft Nucleoid Model

Potential fitted to experimental data

● Experimental data — Simulation result ··· Theoretical probability from the Boltzmann distribution, \(P\left(x\right)\propto e^{-\frac{U\left(x\right)}{k_{B}T}}\)

Soft Nucleoid Results

  • All particles show slightly sub-diffusive behavior

Conclusions

  • The hard nucleoid model is very sensitive to particle size, and went from trapped to diffusive

  • The soft nucleoid showed little sensitivity to particle size, with minimal sub-diffusive behavior

  • A better model for the data shown earlier may require some combination of the two

Activity in the Cell Cytoplasm

  • Metabolic activity shows a strong effect on cellular dynamics

    • Is this a direct effect due to the chemical activity in the cytoplasm, or a secondary effect, e.g. increasing the crowding in the cell?

Wild-type

Inactive metabolism

Colors represent particle size

Previous Work

  • Activity: “the ability of individual units to move actively by gaining kinetic energy from the environment”

  • Applied to flocking and herding of animals, swimming microorganisms, Janus particles [6], and more

Example: Janus Particles

  • Particles that are half coated in platinum are placed in a hydrogen peroxide solution

  • Platinum catalyzes a reaction, driving an osmotic gradient

  • This leads to a molecular motor effect and increased diffusivity

Janus Particle Trajectories in varying concentrations of H2O2

Chemical Activity in Bacteria

How do we model metabolic activity in cells?

  • Events are stochastic and undirected

  • Metabolism in cells is fueled by ATP, which has an energy of \(20 k_B T\)

  • Events are no more rapid than metabolism, and do not increase cell temperature

Simulations

  • Simulate particles in a fluid undergoing Brownian motion

  • Add activity with stochastic kicks of approximately \(20 k_B T\)

  • Vary density and kick frequency

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Simulations

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Without Activity

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With Activity

Results

  • At high frequencies, the kicks raise the temperature of the fluid

  • At low frequencies, the energy is rapidly absorbed by the fluid and there is no effect

  • This holds true over a range of densities and even with \(200 k_B T\) kicks

Conclusion

Activity can only increase diffusion if it is directed, continuous, or at physiologically unfeasible frequencies or energies

Next Steps

Without activity, what effects do we have left?

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Crowding

How does purely exclusive-volume crowding affect dynamics?

Glassy Dynamics: The Ultimate Crowd

  • Glassy dynamics occur at high densities when time-scales for large particle displacements start to diverge

  • Systems with attractive potentials show glassy dynamics, and hard spheres display them in a limited density range

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Glassy Dynamics

Packing Fraction:

\[\phi = \frac{\textrm{volume of particles}}{\textrm{volume of box}}\]

Cooperative Relaxation Model

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Cooperative Relaxation Model

particle movement in a glass requires the cooperative motion of multiple particles, and the size of the region involved in such cooperative motion diverges as the glass transition is approached

What is the evidence for the cooperative relaxation model?

Evidence for Caging

Dynamical Heterogeneities

A common measure for dynamical heterogeneities is \(\alpha_2\):

\[\alpha_{2}\left(\Delta t\right)=\frac{3\left\langle \Delta r\left(\Delta t\right)^{4}\right\rangle }{5\left\langle \Delta r\left(\Delta t\right)^{2}\right\rangle ^{2}}-1\]

\(\alpha_{2} \approx 0\) for Gaussian distributions

\(\alpha_{2} ⪆ 1\) for Bimodal distributions

Dynamical Heterogeneities

\(\alpha_2\) for \(N=100\)

Maximal \(\alpha_2\) for various \(N\)

Unrelaxed simulations are shown with dotted lines.

Conclusions

  • Some evidence for the cooperative relaxation model can be seen in the distribution of step sizes for hard spheres

  • Large values of \(\alpha_2\) are not limited to attractive interactions, and can be seen in hard spheres at high densities

Summary

  • The dynamics of disordered proteins can be accurately modeled with a simple 2-term potential calibrated to experimental data

  • The complicated dynamics inside cells observed in experiments may be linked to the presence of the nucleoid, polydispersity, and crowding (caging) behavior, but active matter is an unlikely candidate

Acknowledgments

  • My Committee!

  • Corey, Mark, and the O’Hern Lab

  • Our collaborators from the Rhoades lab and the Jacobs-Wagner lab

  • The many great teachers I have had

  • My family and my wife

Bibliography

T. Mittag and J. D. Forman-Kay, Current Opinion In Structural Biology 17, 3 (2007).

A. Q. Zhou, C. S. O’Hern, and L. Regan, Biophysical Journal 102, 2345 (2012).

F. Richards, Journal Of Molecular Biology 82, 1 (1974).

V. N. Uversky, J. R. Gillespie, and A. L. Fink, Proteins: Structure, Function, And Bioinformatics 41, 415 (2000).

B. R. Parry, I. V. Surovtsev, M. T. Cabeen, C. S. O’Hern, E. R. Dufresne, and C. Jacobs-Wagner, Cell 156, 183 (2014).

J. R. Howse, R. A. L. Jones, A. J. Ryan, T. Gough, R. Vafabakhsh, and R. Golestanian, Physical Review Letters 99, 048102 (2007).

J. Kyte and R. F. Doolittle, Journal Of Molecular Biology 157, 105 (1982).

O. D. Monera, T. J. Sereda, N. E. Zhou, C. M. Kay, and R. S. Hodges, Journal Of Peptide Science 1, 319 (1995).

Extra Slides

All-Atom and United-Atom Geometry

  • Bond lengths and angles held constant (with a stiff spring)

    • angles and lengths taken from an average over 800 known crystal structures

  • "Atoms" treated as hard-spheres that cannot overlap

    • Repulsive Lennard-Jones potential

2 Carbon atoms with centers at a distance \(r_{ij}\) from each other
\[ V_{ij}^{r}=\begin{cases} 4\epsilon_{r}\left[ \left( \frac{ \sigma^{r}}{r_{ij}} \right)^{12} - \left(\frac{\sigma^{r}}{r_{ij}} \right)^{6}\right] + \epsilon_{r} & r_{ij} < 2^{1/6} \sigma^{r}\\ 0 & r_{ij} > 2^{1/6} \sigma^{r} \end{cases} \]

Coarse-Grained Model Geometry

  • Each monomer represents one residue — many atoms

    • "Bond" lengths and angles

    • Dihedral angles

  • Don’t calibrate to the crystal structures!

  • Calibrated to united-atom and all-atom geometry

Electrostatics

\[V_{ij}^{\textrm{es}}=\frac{1}{4\pi\epsilon_{0}\epsilon}\frac{q_{i}q_{j}}{r_{ij}}e^{ - \frac{r_{ij}}{\ell}}\]

  • \(\epsilon\) is the permittivity of water

  • \(e^{-\frac{r_{ij}}{\ell}}\) gives the Coulomb screening, because we have a 150 mM salt concentration

    • Debye length \(\ell = 9\,\textrm{Å}\)

  • Use partial charges for atoms

Screened Coulomb Potential

Hydrophobicity

\[V_{ij}^{a}=\begin{cases} -\epsilon_{a}\lambda_{ij} & R_{ij}>2^{1/6}\sigma^{a}\\ 4\epsilon_{a}\lambda_{ij}\left[\left(\frac{\sigma^{a}}{R_{ij}}\right)^{12}-\left(\frac{\sigma^{a}}{R_{ij}}\right)^{6}\right] & R_{ij}<2^{1/6}\sigma^{a} \end{cases}\]

  • Lennard-Jones potential

  • \(\epsilon_{a}\) is a parameter we need to determine

  • \(\lambda_{ij}\) is the relative hydrophobicity

  • \(\sigma_{a}=4.8\,\textrm{Å}\) is the average size of a residue

Hydrophobicity Potential

Hydrophobicity Scales

  • Hydrophobicity is a complex interaction that does not map simply onto experimental measurements

  • Several groups have devised separate scales for evaluating hydrophobicity

Hydrophobicity per Residue

Hydrophobicity Models

Scales1- Kyte-Doolittle [7]2- Monera [8]3- Average of 7 scales
Mixing Rule-1 Arithmetic mean \(h_{ij}=\frac{h_{i}+h_{j}}{2}\)-2 Geometric mean \(h_{ij}=\sqrt{h_{i} h_{j}}\)-3 Maximum \(h_{ij}=\max(h_{i},h_{j})\)
ProteinRed: αSBlue: βSGreen: γSPurple: ProTαOrange: MAPT