TSA3 Struggle Topics - MD and Protein interactions

Why is energy minimisation before MD important?

Energy minimisation is required to reach the most stable, non-strayed configuration of a molecule (eg. a protein).

If this process is not carried out, the structure you are doing MD with will be strained. This will not reflect its lowest energy, physiological configuration and so will likely cause MD calculations to fail.

What are 2 examples of different methods of energy minimisation in MD?

• Steepest descents

• Conjugate gradients

How does steepest descents contribute to finding the energy minima in MD?

Steepest descents looks for local minima using gradients.

At any point, the first derivative of a function is found. The negative gradient at this point Is determined and used to go downhill.

What methods of steepest descents can be used to apply the negative gradient in order to determine a local energy minima?

• Arbitrary Steps

• Line Search

How does arbitary steps attempt to reach an energy minima? What is/isnt good about this method?

This method moves forwards and backwards (along the defined negative function of the gradient). At each new position, the energy is compared against the energy of the previous position to determine whether this point is closer to an energy minima or not.

However, because this is random and arbitrary, this method can be slow and inefficient

How does a line search attempt to reach an energy minima? What is/isnt good about this method?

Linesearches fit a quadratic equation to the current region of the energy landscape. this is used to move towards the local minima.

This is more efficient than arbitrary steps as it actively moves towards a minimum.

Why may you not use steepest descents in energy minimisation? What might you switch to?

You may choose not to use steepest descents when you are close to the energy minima. This is because when close to the minima, steepest descents may fail to actually reach the minima, and will instead zig zag around it.

May choose to switch to a different method, such as conjugate gradients in order to reach the actual minima

Why are there different methods of energy minimsation?

• Each method varies in efficiency relative to how close to the minima the structure is

• Therefore, we need different methods to reach different points, and then we can switch to a different method that will allow us to get closer to the energy minima

What is the leapfrog integrator?

An algorithm used to calculate the changes in positions and velocities of atoms over time.

What is special about when positions and velocities are calculated using the leapfrog integrator?

The positions and velocities are calculated out of phase with each other by half a time step. this means that they are never simeltaneously known.

Explain the first step of the leapfrog integrator at time t.

• At time t, the velocity from half a time step before (t - ½Δt) is known. The current position of all atoms is also known

• The positions of all the atoms is used to determine the potential energy of the atoms.

• This potential energy is used to determine the forces acting on the atoms via the following relationship: F(t) = -dU/dr

• This force is related to acceleration, allowing determination of the acceleration of atoms at time t ia the following relationship: a(t) = F(t)/m

Overall, what happens in the first step of using the leapfrog integrator?

Positions are used to determine forces, which are used to determine acceleration

What happens in the leapfrog integrator once accelerations are known?

Once accelerations are known, the velocity at half a time step forward is calculated. This is possible because we assume acceleration is constant

This new velocity is used to update the positions of atoms, moving them a time step forward.

We then repeat this process

what are velocities used to determine in the leapfrog integrator?

velocities are used to determine the updated positions of atoms

overall, what is the process of the leapfrog integrator?

initial positions are used to determine the force on atoms

this force is used to determine the acceleration of atoms

acceleration of atoms is used to update the velocity at a time step forward than the previous one, and this is half a time step forward than the known positions

this new velocity is used to determine the new positions of atoms, moving their positions a complete time step forward.

Why is block averaging of MD trajectories needed, rather than just averaging a set of values from successive frames?

If frames are treated independently the uncertainty between them will look very small as they are correlated.

block averaging overcomes this by dividing the trajectory into defined blocks, averaging these and allowing comparisons to be made with real uncertainties

explain how block averaging works

• dives the trajectory into blocks, with a defined number of M blocks. each of these blocks has the same length, L

• Each block contains many correlated frames, however different blocks will not, and so can be treated independently

• We determine the average of each block - this effectively coarse grains the trajectory into block averages, rather than individual frames

• The variation between the block averages is compared and used to determine the SEM via the number of blocks.

why must blocks be the correct length in block averaging? what is done to ensure this correct length?

• if blocks are too small, blocks will be strongly correlated, and so can no longer be treated independently

• to ensure this correct length, we must make sure that the first frame of the block cannot predict the last frame - this occurs when the block length is higher than the correlation time

  • this ensures that block averages behave as independent observations

what is a systematic error?

a systematic error is where a value given by a system differs from thet given by the real system

briefly, what gives rise to systematic error?

systematic error is a result of approximating a system in the simulation. these systems can never be accurately approximated, and so errors due to the underlying representation can cause discrepancies

These are systematic because they arise from how the simulation/model is constructed and calculated, rather than from limited statistical sampling.

what parts of a simulation may cause systematic errors to arise in MD?

• Algorithmic/numerical errors – approximations or limitations in the mathematical algorithms used.

• Force-field limitations – certain interactions may be missing or not represented accurately, e.g. cation–π interactions.

• Force-field parameterisation errors – interaction parameters may be inaccurate, particularly for difficult systems such as transition metals.

• Finite machine precision / rounding errors – computers can only represent numbers to finite precision, so small rounding errors can accumulate over many timesteps.

what is statistical error in MD simulations? what are these a result of?

• statistical errors are when an average value In a simulation of a system does not reflect that of the real value of a system

• result of under sampling - simulations only sample a small region of the energy landscape, and so will exclude regions, preventing these from being able to contribute to averages

how can systematic errors in MD simulations be corrected/reduced?

• Using forcefields that represent the type of study being used

• validating results against experimental data

• using more sophisticated models that represent in vivo systems - eg. could use polarisable forcefields to reflect the changes in electrostatics that a given charge will experience

cannot be corrected by increasing simulation time

how can statistical errors in MD simulations be corrected/reduced?

• running a longer simulation is the only way to correct for statistical errors. this allows the simulation to sample more phase space, this means that a larger region of the energy landscape is sampled, and so more real values contribute to the mean, making the observed, simulated mean closer to the actual mean of the system

Why do we need averages when analysing MD simulations?

• MD simulations are trajectories, which give a view of the different configurations of a system over time.

• However, when measuring values, including energy, temperature etc, we need values that represent the entire ensemble that the system has sampled, and so singular frames of the trajectory may not be representive of this. therefore, trajectory averages are used

why use block averages when determining trajectory averages of MD simulations?

• frames of the trajectory related closely by time will be strongly correlated, and therefore using them to determine an average can be computationally intensive and redundant

• instead, block averages can be used to reduce computational intensity - we can use different methods to select how we determine these block averages

what are the different types of block averages that can be used when analysing MD trajectories?

• stratified systematic sampling

• stratified random sampling

• coarse grained averaging

how does stratified systematic sampling choose a block average?

• in stratified systematic sampling, one frame from each block is selected. these frames are selected at a defined position throughout the blocks

what advantage does stratified systematic sampling offer?

• selecting the averages at a defined position in each block gives configurations that are evenly distributed across the entire trajectory

what must you be careful of in regards to frames when using stratified systematic averages?

• frames must be sufficiently separated (and therefore blocks must be sufficiently long). otherwise, this could lead to correlated frames being selected, which would be redundant

how does stratified random sampling choose a block average?

• in stratified random sampling, equal length blocks are used, but one frame from each block is selected randomly to be used as the average

what advantage does stratified random sampling offer?

• the use of blocks ensures that the whole trajectory is sampled - if a completely random approach was used, this could mean that averages are only selected from one region of the trajectory, which could prevent reliable averages from being determined

• the random sampling of a frame in each bock avoids sampling bias that could be produced by periodicity used in systematic sampling

what disadvantage does stratified random sampling offer?

• because there is little bias in the frames selected per block, this could result in correlated frames being selected

how does coarse grained block averaging choose a block average?

• this method uses blocks, but all of the values within each block are averaged to produce a block average

• these block averages are then averged to produce a trajectory average

what is coarse grained block averaging good for?

• determining values of thermodynamic properties that display inherent fluctuation.

• for example. energy, temperature, pressure, density

what is stratfied block averaging good for?

• good for structural analysis

what is the potential of mean force?

• the PMF is a measured value that describes the change in free energy along a gien coordinate.

• this allows us to view and analyse how thermodynamically favourable different positions along a pathway are

what is the problem when trying to determine PMF?

• MD simulations are timescale limited, this means that spontaneous events might not be seen, as they often have large energetic barriers associated with them

how does umbrella sampling allow spontaneous events/positions to be sampled to allow PMF to be determined?

• umbrella sampling uses bias (via harmonic restraints) to force a molecule to sample defined positions of an energy diagram. this allows us to determine the potential of mean force

how do we do umbrella sampling?

• we force the molecule to sample defined positions along a reaction coordinate

• this is done by running individual simulations at successive positions along the axis

• we restrain the moelcule to these positions to ensure it samples them using harmonic restraints

• furthetrmore, these positional windows must overlap to allow the free energy diagrm to be constructed

how can results from umbrella sampling be used to get PMF?

becuase we know the biasing potentials applied to restrain molecules to defined positions along a reaction coordinate, these can be removed to produce an unbiased ree energy profile, the PMF

how are umbrella sampling and PMF related?

umbrella sampling is a method used to calculate PMF

what does steered MD do?

steered MD moves a system towards a known desired state by applying an external pulling force

how can steered MD use collective variables to drive a system towards a defined state?

collective varaibles can be used to drive a system. for example, you could drive movement towards a desired state by measuring a decrease in RMSD

how can free energy difference between states be determined in steered MD?

• you can use the work done from many repeated steered MD runs and the jarzynski equaltity to find the free energy diffeence between states

what problem does metadynamics solve? how?

• in an MD run, simulations can get trapped inenergy minima, and therefore fail to sample more states - this wastes cmputatonal power

• metadynamics can be used to add bias to regions that have already been sampled, allowing the system to escape these and sample more reigions

how does metadynamics improve sampling?

• artificial bias is applied to already samped regions. becuase this increases the energy of these regions, these are no longer favourable to ocupy, alowing th system to overcome energetic barriers to explore the energy landscape and more states

what does biasing do in metadynamics?

in metadynamics, bias flattens the effective energy landscape, this makes state transitions more favourable, promoting exploration of new regions of conformational space

how can we get a free energy landscape in metadynamics?

in metadynamics, we can get the free energy landscape becuase we now what the bias we have added looks like, and so use this to give us information about the free energy landscape

What is T2?

T2 is the transverse relaxation time - this explains how quickly transverse magnetisation in the XY plane loose phase coherence and decays, making it undetectable

Why can T2 not be determined directlly from NMR experiments?

• We cannot measure true transverse decay, and so cannot fit it to an exponential curve to get T2

• This is becuase Bo is not perfectly homogenous, ans so different nuclei experience different Blocal, causing them to precess at different frequences, increasing dephasing and broadening the NMR signal

what must we do to get T2?

we need to remove the dephasing of spins caused by field inhomogeneity

explain the process of the spin echo approach

• firstly, we apply a 90º RF pulse to the sample - this flips nuclei into the xy plane, and spins precess

• not all spins precess at the same frequency due to differences in Blocal experienced, this causes them to fan out in the transverse plane and begin o dephase

  • some of this dephasing is inherent (due to T2 relaxation), but some is due to the field inhomogeneity

• After the spins have precessed for a period of tau (τ), a 180º RF pulse is applied, this again flips the spins

• we then wait for another period of tau (τ). during this waiting period, the reversible dephasing is undone and spins come back together, refocusing and forming a spin echo after a period of 2τ

• we measure the intensity of this spin echo

what experiment do we use to get T2

spin echo

in the spin echo experiment, after what period to spins refocus and echo?

t = 2τ

how do we use spin echo experiments to get T2

we use the intensity of the spin echos with their related τ’s to plot a graph of intensity vs t

this is fit to an exponential curve, and this is used to get T2

what exponential equation are the intensity values and τ fit to in the spin echo experiment?

I(t) = I(o)e^-t/T2

what is T1?

T1 is the longitudinal relaxation time - this is how quickly net magnetisation in z returns to its equilibrium value and aligns with Bo

why cant we get T1 directly from NMR experiments?

We cannot get T1 directly becuase magnetisation along z is not detected in NMR experiments

briefly, what do we do in T1 spin inversion experiments?

at defined timepoints of relazation, we flip magnetisation into xy and measure intensity

describe the spin inversion experiment to get T1

• we start at equilibrium, the net magnetisation is aligned with Bo (and so is in +z)

• we flip the magnetisation to align with -z using a 180º RF plse

• Duriing a waiting period of time τ, the longitufinal magnetisation begins to relax towards the equilibrijm via an exponential relationshop

• After the defined period of τ, we apply a 90º RF pulse, this flips the magnetsation into xy - now it can be detected

  • here, we record the intensity of the signal

• We repeat this using different lengths of τ - each gives a signal intensity reflective of the recovery of longitufinal magnetisation

• this is used to fit a graph of I and τ to get T1

what does the spin inversion experiment aim to get?

T1

what equation is used in T1 spin inversion experiments?

I(τ) = Io(1-2e^-(τ/T1) )

why can we plot the exponential decay in spin inversion experiments as so?

we can substitute Mz(τ) for I(τ) becuase the two are proportional, and so intensity can directly be used to get T1

what are the NMR techniques used to probe fast motions?

15N T1, 15N T2, 1H-15N NOEs

What do T1, T2 and NOE depend on?

spectral density

what is spectral density?

a parameter describing the amplitude of molecular fluctuations at different frequencies

What causes the local magnetic-field fluctuations that drive NMR relaxation?

Molecular motion, including Brownian tumbling and internal motion, which changes the magnetic interactions between nuclei over time.

What is the rotational correlation time, τc?

The characteristic time over which a molecule loses memory of its initial orientation.

What does a small τc mean?

fast molecular tumbling

What does a large τc mean?

slow molecular tumbling

How does molecular size generally affect τc?

larger molecules tumble more slowly → larger τc.

Can local internal motion affect the effective correlation time of a bond?

Yes. Internal flexibility can make a particular bond vector reorient faster than would be expected from whole-molecule tumbling alone and therefore can effect τc

Why are backbone 1H-15N NOEs useful for studying motion?

Because the H–N distance is fixed by the covalent bond, so changes in NOE mainly reflect motion rather than changes in distance.

What does a high positive heteronuclear NOE generally indicate?

More restricted, rigid motion of the N–H bond.

How does T2 change with τc

• changes monotonically

• small tauC = longer T2

How does T1 change with τc?

• non monotonic relationship

• graph has a minima

Why are T1, T2, and heteronuclear NOE interpreted together rather than separately?

each depends differently on the spectral density, so combining them gives a better description of molecular motion

What is the purpose of model-free analysis?

To use the relaxation data to describe overall tumbling and local internal motion without requiring a detailed structural model for the motion

What does the order parameter S^2 describe?

The amplitude of local internal motion.

What does the internal correlation time τe describe?

how quickly local internal motion occurs

What is the difference between S² and τe?

S² describes how much the bond moves, whereas τe describes how fast that internal motion occurs.

What is the difference between τc and τe?

τc describes whole-molecule tumbling, while τe describes local internal motion.

what do cpmg experiments detect? why?

these experiments detect conformational exchnage

this is possible because the echnage between states with different chemical shifts contributes to transverse relaxation

how do cpmg experiments allow us to understand how chemical exchange processes contrubute to transverse relaxation?

by varying the frequency of 180º refocusing pulses, we supress the contrubution of the chemical exchnage.

we can then fit the dependence of the effective transverse relaxation rate (1/T2) agaisnt CMPG frequency to determine exchnage rate, state populations and chemical shift differences between exchnaging conformations

what information can we gain from CPMG relaxation dispersion experiments?

exchange rate, state populations, and chemical-shift differences between the exchanging conformations

what is chemical exchange?

• proteins, or regions of proteins, are often dynamic - this means they can adopt different conformations. in different conformations, there will be different NMR chemical shifts. this is becuase nuclei will experience different chemical environments in these different chemical shifts

• the difference in chemical shifts between these environments is Δ⍵

what is the overall exchnage rate between two protein conformations?

kex = kab + kba

why is the overall exhcnage rate important in chemical exhcnage?

• the overall exchnage rate, and its magnitude relative to Δ⍵ dictates the shape of the NMR spectra

what happens to the NMR spectra of a protein in slow exchange?

• In slow echnage, Δ⍵ » kex.

• this means that NMR can distinguish each chemcial environment, as the protein occupies each enough

• this produces 2 distinct peaks, reflective of the two protein conformations. furthermore, the intensities of each of these peaks reflects the populations of each state

what happens to the NMR spectra of a protein in fast exchange?

• In slow echnage, Δ⍵ « kex.

• this means that NMR cannot distinguish each chemcial environment, as the protein does not occupy each enough

• this produces 1 intermediate peak, between the two typical peaks we would see in the slow exchange regime. this position is dependent on the populations of each state being exchanged between

what happens to the NMR spectra of a protein in intermediate exchange?

• In intermediate echnage, Δ⍵ ~ kex.

• this produces a broad peak, reducing t2 due to enhanced transverse relaxation

• this occurs because the molecules transitioning alters the precession frequency. this allows them to accumulate phase, leading to a loss of phase coherence more quickly

what is force field parameterisation?

choosing the correct numerical values in a force field to ensure that it reproduces real molecular behaviour

how are force fields parameterised?

• we fit forcefield values using real experimental data and QM calculations

Explain what fluorescence polarisation is in the context of fluorescence anisotropy experiments

• When fluorophores emit photons after excitation, the absorption and emission are direction dependent.

  • this is because fluorophores have an electronic transition dipole moment, meaning that absorption is most efficient when the electronic transition dipole is aligned with the electric field of the incoming excitation light

• Therefore for a solution of fluorophores being excited with polarised light, only a small subset will be preferentially excited - these are the ones with absorption transition dipoles parallel to the excitation light.

What is photoselection?

• for a solution of fluorophores being excited with polarised light, only a small subset will be preferentially excited - these are the ones with absorption transition dipoles parallel to the excitation light.

What is fluorescence anisotropy?

When fluorophores are excited with polarised light, only those with electronic transition dipole moments aligned with the excitation light will be excited.

However, in the fluorescence lifetime (the time between initial excitation and emission) molecules can undergo rotational brownian motion. This means that their emission direction can change. This change in direction (due to rotatio) can be impacted by binding partners, this is due to the dependence of fluorescence anisotropy on the rotational correlation time.

describe that can happen to the emission direction of fluorescnt ligands that are free in fluorescence anisotropy experiments?

• small molecules will rapidly tumble, and so can rotate substantially in the fluorescence lifetime.

• this means that their emission direction can become random

• this produces low fluorescence anisotropy

describe that can happen to the emission direction of fluorescent ligands that are protein bound in fluorescence anisotropy experiments?

• Larger molecules will tumble more slowly, and so rotate less in the fluorescence life time

• this means that their emission direction is less random, and the emitted fluorescence will retain more of the original polarisation

• this produces high fluorescence anisotropy

what is the equation for the dependence of anisotropy on fluorescence life time and rotational correlation time?

r = r_o / (1 + τ/θ)

where r = anisotropy, tau = fluorescence lifetime and theta = rotational correlation time