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
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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 model
and arrive at realistic results?
Previous Work
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Methods for α-Synuclein
Molecular Dynamics Simulations
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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 |
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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) | Lennard-Jones Repulsive (WCA) | \(\sigma=4.8\,Å\), from \(R_{g}\) of residues | |
Building a Model: Long-Range Interactions
| Electrostatics | Hydrophobicity |
|---|---|
\[V_{ij}^{\textrm{es}}=\frac{1}{4\pi\epsilon_{0}\epsilon}\frac{q_{i}q_{j}}{r_{ij}}e^{-\frac{r_{ij}}{\ell}}\]
| \[V_{ij}^{a} \propto\left(\frac{\sigma^{a}}{R_{ij}}\right)^{12}-\left(\frac{\sigma^{a}}{R_{ij}}\right)^{6} R_{ij}>2^{\frac{1}{6}}\sigma^{a}\]
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Full Model
Results for α-Synuclein
All-Atom Geometry
All-Atom | PDB Structures |
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Zhou et al. [2] provided atom sizes calibrated to a hard sphere model
United-Atom Geometry
United-Atom | PDB Structures |
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Richards et al. [3] provided atom sizes calibrated to calculate packing densities; we multiplied by 0.9
Radius of Gyration (\(R_{g}\))
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\[\alpha=\frac{\textrm{Hydrophobicity Strength}}{\textrm{Electrostatic Strength}}\] |
smFRET
Single-Molecule Förster Resonance Energy Transfer
smFRET of α-synuclein
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smFRET Comparison (United-Atom)
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\[F_{\textrm{eff}}=\left\langle \frac{1}{1+\left(\frac{R_{ij}}{R_{0}}\right)^{6}}\right\rangle\] |
smFRET Comparison (Coarse-Grained)
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\[F_{\textrm{eff}}=\left\langle \frac{1}{1+\left(\frac{R_{ij}}{R_{0}}\right)^{6}}\right\rangle\] |
smFRET Comparison
United-Atom | Coarse-Grained |
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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 |
Uversky et al. [4] showed that charge and hydrophobicity
were predictors of disordered proteinsThey drew a line at \(Q=2.785H-1.151\)
smFRET Comparisons
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Radius of Gyration (\(R_g\))
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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 |
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Radius of gyration of 5 proteins | Scaling of partial \(R_g\) with chemical distance |
Radius of Gyration Scaling
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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 proteinsWe 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 | |
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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 |
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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
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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
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Potential fitted to experimental data | ● Experimental data |
Soft Nucleoid Results
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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?
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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
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Simulations
Without Activity | With Activity |
Results
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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?
Crowding
How does purely exclusive-volume crowding affect dynamics?
Glassy Dynamics: The Ultimate Crowd
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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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What is the evidence for the cooperative relaxation model?
Evidence for Caging
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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 \(N\) |
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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).
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
|  2 Carbon atoms with centers at a distance \(r_{ij}\) from each other
\[
V_{ij}^{r}=\begin{cases}
4\epsilon_{r}\left[ \left( \frac{ \sigma^{r}}{r_{ij}} \right)^{12} - \left(\frac{\sigma^{r}}{r_{ij}} \right)^{6}\right] + \epsilon_{r} & r_{ij} < 2^{1/6} \sigma^{r}\\
0 & r_{ij} > 2^{1/6} \sigma^{r}
\end{cases}
\]
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Coarse-Grained Model Geometry
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Electrostatics
\[V_{ij}^{\textrm{es}}=\frac{1}{4\pi\epsilon_{0}\epsilon}\frac{q_{i}q_{j}}{r_{ij}}e^{ - \frac{r_{ij}}{\ell}}\]
|  Screened Coulomb Potential |
Hydrophobicity
\[V_{ij}^{a}=\begin{cases}
-\epsilon_{a}\lambda_{ij} & R_{ij}>2^{1/6}\sigma^{a}\\
4\epsilon_{a}\lambda_{ij}\left[\left(\frac{\sigma^{a}}{R_{ij}}\right)^{12}-\left(\frac{\sigma^{a}}{R_{ij}}\right)^{6}\right] & R_{ij}<2^{1/6}\sigma^{a}
\end{cases}\]
|  Hydrophobicity Potential |
Hydrophobicity Scales
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Hydrophobicity per Residue |
Hydrophobicity Models
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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 \(h_{ij}=\max(h_{i},h_{j})\) | Protein Red: αS Blue: βS Green: γS Purple: ProTα Orange: MAPT | |||