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One metaphor that I have found to explain how proteins fold so quickly to a native shape is that of the blind golfer.

I have made a video to illustrate this metaphor. This shows how the shape of a slope can determine precisely where individual elements will end up.

The video represents many zero-dimensional points each following its own one-dimensional path down a three-dimensional slope to end up in a two-dimensional arrangement. Or almost: the one-dimensional paths are curves in three-dimensional space, the final resting surface is not flat, and one marble sits on top of three others.) The movement takes place through the fourth dimension of time.

Proteins form three-dimensional shapes, so the "slope" that their components slide down must be in an extra dimension.

Atoms have been observed to jump through a crystal lattice, without any indication that, in travelling from their start point to their end point, they pass through the three-dimensional space that we can observe. Is it conceivable that the component parts of a protein move through an extra dimension as the protein "folds"?

There are very simple protein chains whose native folded shape is well known. Is there any data available to suggest what path the different molecules in the protein follow during the folding process? Or do they appear to "jump"?

I imagine that it would be possible to create a mathematical model in n-dimensions to describe the simplest "slope" that the protein molecules could follow as the protein collapses into its native state.

What mathematical work is currently being done in this area?

jeff101's picture
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Proteins have many coordinates:

In molecular dynamics simulations, every atom in a protein has xyz position and velocity coordinates, and all these coordinates can vary with time. For N atoms, this gives 6N coordinates that can all vary with time.

Often other coordinate systems are used; for example, phi,psi dihedral angles for the protein backbone and other angles for the protein sidechains. These coordinates can vary with time as well.

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Re: Proteins have many coordinates:

That's 3N. You can't rotate a point (and atoms are generally modelled as pointlike particles).

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Why 6N coordinates for N atoms?

I think 6N coordinates is more true than 3N coordinates for N atoms.
If you consider just one atom in space, you have 3 coordinates x,y,z
for its position and 3 coordinates vx,vy,vz for its velocity. The
future behavior of this atom depends on all 6 of these coordinates.

For example, if the atom were in a potential well with the
potential energy U=(1/2)k(x^2 + y^2 + z^2), it would feel the
forces Fx,Fy,Fz in the x,y,z directions, respectively. Following the
notation in http://fold.it/portal/node/2001037#comment-30906 below,
we would have:

Fx = m*ax = m*dvx/dt = m*d2x/dt2 = -dU/dx = -k*x
Fy = m*ay = m*dvy/dt = m*d2y/dt2 = -dU/dy = -k*y
Fz = m*az = m*dvz/dt = m*d2z/dt2 = -dU/dz = -k*z

These all obey Hooke's Law for springs and make
3 simple harmonic oscillators, one in each of the x,y,z directions.
Their solutions versus time are:

x=Ax*sin(w*t+px)   vx=dx/dt=w*Ax*cos(w*t+px)    
y=Ay*sin(w*t+py)   vy=dy/dt=w*Ay*cos(w*t+py) 
z=Az*sin(w*t+pz)   vz=dz/dt=w*Az*cos(w*t+pz)  

ax=dvx/dt=d2x/dt2=-w*w*Ax*sin(w*t+px)=-w*w*x
ay=dvy/dt=d2y/dt2=-w*w*Ay*sin(w*t+py)=-w*w*y 
az=dvz/dt=d2z/dt2=-w*w*Az*sin(w*t+pz)=-w*w*z 

where Ax,Ay,Az are the oscillation amplitudes in each direction,
w is the angular frequency (it obeys w^2=k/m) in all directions,
and px,py,pz are the phases of each oscillation at time 0.

If the atom began at time t=0 with x,y,z all equal to 0,
you might think it was at equilibrium. After all, it would be
at the minimum potential energy (U=0), and the forces Fx,Fy,Fz
on it would all be 0. Nevertheless, its future behavior depends
not only on the positions x,y,z but also on the velocities vx,vy,vz
at time 0.

For example, x could have Ax=0 so that x=0 vx=0 and ax=0 hold for all time t.
Next, y could have Ay=3 and py=0 so that y would oscillate between -3 and +3
versus time, giving y=3 when w*t=pi/2 radians (90 degrees). Finally, z could
have Az=5 and pz=pi radians (180 degrees) so that z would oscillate between
-5 and +5 versus time, giving z= -5 when w*t=pi/2 radians (90 degrees).

There are many other ways to have x,y,z all equal to 0 at time 0.

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Equations of motion:

If we consider just the motion of one atom in the x-direction, we have:

x = position of the atom in the x-direction

vx = dx/dt = velocity of the atom in the x-direction
   = 1st derivative of the position x

ax = dvx/dt = d2x/dt2 = acceleration of the atom in the x-direction
   = 1st derivative of the velocity vx
   = 2nd derivative of the position x.

We also have Fx = m * ax
(the force Fx in the x-direction equals the mass m of the atom times its acceleration ax in the x-direction).

Finally, the many interactions between atoms (attractions of opposite-sign charges, repulsions of
same-sign charges, preferred bond-lengths and angles, etc.) give a potential energy U that depends
on the xyz positions of all the atoms. Fx, the force in the x-direction on an atom, is related to U by
Fx = -dU/dx, where dU/dx is the slope of the potential energy U when plotted vs x at a particular value of x.
The minus sign in Fx = -dU/dx makes the force Fx point toward lower potential energy U.

Note that each atom will have its own expression for U,
so most atoms will feel different forces from each other.
Each atom's U is like the elevation of a rugged mountain,
and the force the atom feels at a particular point is always
aimed downhill. If you drew contour lines on the mountainside
with equal elevation (equal potential energy U), the lines of
force would always be perpendicular to the lines of equal
elevation or equal potential and would always point downhill.

Similar to the above, forces in the y and z-directions obey:
Fy = m * ay and Fy = -dU/dy
Fz = m * az and Fz = -dU/dz
where:
ay = dvy/dt = d2y/dt2 and vy = dy/dt and
az = dvz/dt = d2z/dt2 and vz = dz/dt.

At equilibrium, the velocities, accelerations, and forces all go to zero,
and each atom should be at a local minimum in its U (and sometimes at a
global minimum, the deepest possible valley, in its U).

I hope this helps!

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Yes! (And no, a little bit.)

Yes, higher-dimensional parameter spaces are essential to modelling protein folding. A conformation of a protein is best thought of as a point in a higher-dimensional space. Given x, y, and z-coordinates of N atoms in the system, the parameter space describing the system is 3N-dimensional (e.g. 3000-dimensional if we have 1000 atoms). It's inconvenient to work with a space of such large dimensionality, though, especially when most of that space is inaccessible, representing impossible conformations with ridiculously stretched bonds or horrible bond angles or whatnot. So we tend to confine ourselves to the curved subspace of that space in which all bond angles and bond lengths are ideal: torsion space. Torsion space is an M-dimensional curved manifold in the larger 3N-dimensional space (where M << 3N). By way of analogy, imagine you wanted to give someone the coordinates of a city on Earth. You could specify three coordinates corresponding to a point in 3D space, but to a good approximation, every city lies on the surface of a sphere, a curved 2-dimensional subspace of the overall three-dimensional space. So you can get away with specifying a longitude and a latitude. Similarly, if I wanted to specify the conformation of a protein with N atoms precisely, I'd have to give you 3N numbers (the x, y, and z coordinates of every atom) -- but I probably could get away with just telling you M main-chain and side-chain torsion angles, and let you assume that all bond angles and bond lenghts are ideal.

The wiggle tool in Foldit (what we call the minimizer in Rosetta) is actually calculating derivatives of the energy in the M-dimensional torsion space -- or rather, an M-vector called the gradient of the energy, which is equivalent to the slope of a line in one dimension. Since the gradient points "uphill", and since we want to minimize the energy, we start marching in the opposite direction -- "downhill" until we find we're going "uphill" again, backtrack to the lowest point encountered, then re-calculate the gradient and continue to march "downhill" until we find the nearest local minimum in the M-dimensional space. Thus, by doing some math in higher-dimensional spaces, we're able to quickly and efficiently find the nearest local minimum in the energy landscape to a starting conformation.

Now, the dimensionality of the conformation space, or of any other parameter space used to describe a physical system, should never be confused with actual spatial dimensions; nor should the extra spatial dimensions posited to explain certain sub-nuclear phenomena be conflated with what goes on on the much, much larger scale of protein folding. The physics of protein folding is fairly well understood, and the systems involved are large enough and massive enough to be approximated fairly well by Newtonian physics (with some quantum mechanics-derived potentials) -- all in three spatial dimensions. It's the parameters describing the conformation that can be thought of as defining a higher-dimensional space; the protein itself is a strictly three-dimensional object. (To illustrate this, if I give you my height, weight, shoe size, and collar size, I've given you four parameters describing me -- and haven't come close to exhaustively listing my parameters. You could put those on four mutually-orthogonal axes to define a four-dimensional space, and plot points in that four-space for different people. That doesn't change the fact that I'm a three-dimensional object.)

Note too that the "jumping" that you describe for some physical systems (quantum tunnelling, it sounds like) is unrelated to higher spatial dimensions. As far as we can tell, proteins are too large, too massive, and too warm for tunnelling to have a significant effect on their kinetics of folding. (Still, this was something I wondered about a bit when I was a grad student working on protein folding kinetics.)

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For more on higher-dimensional folding funnels

For more on higher-dimensional folding funnels, I recommend looking up some of Hue Sun Chan and Ken Dill's papers from the 1990s. In particular, their 1997 article, "From Levinthal to pathways to funnels" (Nature Structural Biology 4(1):10-19, available here: http://faculty.uml.edu/vbarsegov/teaching/bioinformatics/papers/dill.pdf ) gives a pretty good overview of the higher-dimensional parameter-space landscape view of protein folding. This seems kind of similar to the analogy that you're using in your video -- though note the ruggedness of the actual landscape, and the many kinetic traps (local minimia) that a protein falls into along the way to a folded state.

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