Energetics of protein structures

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					     Energetics of protein structures




     Energetics of protein structures

• Molecular Mechanics force fields

• Implicit solvent

• Statistical potentials




     Energetics of protein structures

• Molecular Mechanics force fields

• Implicit solvent

• Statistical potentials




                                        1
                    What is an atom?

• Classical mechanics: a solid object

• Defined by its position (x,y,z), its shape (usually a
  ball) and its mass

• May carry an electric charge (positive or negative),
  usually partial (less than an electron)




Example of atom definitions: CHARMM


 MASS   20   C      12.01100   C   !   carbonyl C, peptide backbone
 MASS   21   CA     12.01100   C   !   aromatic C
 MASS   22   CT1    12.01100   C   !   aliphatic sp3 C for CH
 MASS   23   CT2    12.01100   C   !   aliphatic sp3 C for CH2
 MASS   24   CT3    12.01100   C   !   aliphatic sp3 C for CH3
 MASS   25   CPH1   12.01100   C   !   his CG and CD2 carbons
 MASS   26   CPH2   12.01100   C   !   his CE1 carbon
 MASS   27   CPT    12.01100   C   !   trp C between rings
 MASS   28   CY     12.01100   C   !   TRP C in pyrrole ring




Example of residue definition: CHARMM

   RESI ALA            0.00
   GROUP
   ATOM N    NH1      -0.47    !          |
   ATOM HN   H         0.31    !       HN-N
   ATOM CA   CT1       0.07    !          |     HB1
   ATOM HA   HB        0.09    !          |    /
   GROUP                       !       HA-CA--CB-HB2
   ATOM CB   CT3      -0.27    !          |    \
   ATOM HB1 HA         0.09    !          |     HB3
   ATOM HB2 HA         0.09    !        O=C
   ATOM HB3 HA         0.09    !          |
   GROUP                       !
   ATOM C    C         0.51
   ATOM O    O        -0.51
   BOND CB CA N     HN N CA
   BOND C CA C      +N CA HA       CB HB1    CB HB2    CB HB3
   DOUBLE O C




                                                                      2
                            Atomic interactions

                                                Non-bonded
  Torsion angles
                                                pair
  Are 4-body




                                                Angles                  Bonds
                                                Are 3-body              Are 2-body




                           Forces between atoms
  Strong bonded interactions
        b


                           U = K (b ! b0 ) 2        All chemical bonds




        θ
                           U = K (! " ! 0 ) 2       Angle between chemical bonds




                                                    Preferred conformations for
        φ                                           Torsion angles:
                                                      - ω angle of the main chain
                                                      - χ angles of the sidechains
                      U = K (1 " cos(n! ))                  (aromatic, …)




      Forces between atoms: vdW interactions
            r
                                                          1/r12




Lennard-Jones potential                                           Rij

                      12
                 && R #               6                                        1/r6
                             &R #         #
ELJ ( r ) = ( ij $ $ ij ! ' 2$ ij !       !
                 $$ r !      $ r !        !
                 %% "        % "          "


        Ri + R j
Rij =              ; ! ij = ! i! j
            2




                                                                                      3
    Example: LJ parameters in CHARMM




        Forces between atoms: Electrostatics
                    interactions
              r



                                            Coulomb potential



                                                               1 qi q j
         qi          qj

                                         E (r) =
                                                             4"! 0! r




Some Common force fields in Computational Biology


  ENCAD (Michael Levitt, Stanford)

  AMBER (Peter Kollman, UCSF; David Case, Scripps)

  CHARMM (Martin Karplus, Harvard)

  OPLS (Bill Jorgensen, Yale)

  MM2/MM3/MM4 (Norman Allinger, U. Georgia)

  ECEPP (Harold Scheraga, Cornell)

  GROMOS (Van Gunsteren, ETH, Zurich)



 Michael Levitt. The birth of computational structural biology. Nature Structural Biology, 8, 392-393 (2001)




                                                                                                               4
              Energetics of protein structures

   • Molecular Mechanics force fields

   • Implicit solvent

   • Statistical potentials




                                                 Solvent
                                        Explicit or Implicit ?




                        Potential of mean force
A protein in solution occupies a conformation X with probability:

                                    U (X ,Y )
                                "                            X: coordinates of the atoms
                            e          kT
                                                                of the protein
      P( X , Y ) =            U (X ,Y )
                            "
                      !!e        kT
                                          dXdY               Y: coordinates of the atoms
                                                                of the solvent


The potential energy U can be decomposed as:                 UP(X): protein-protein interactions

       U ( X , Y ) = U P ( X ) + U S (Y ) + U PS ( X , Y )   US(X): solvent-solvent interactions

                                                             UPS(X,Y): protein-solvent
                                                                       interactions




                                                                                                   5
                  Potential of mean force
We study the protein’s behavior, not the solvent:


               PP ( X ) = ! P( X , Y )dY

PP (X) is expressed as a function of X only through the definition:
                                           WT ( X )
                                       "
                                   e         kT
                  PP ( X ) =            WT ( X )
                                    "
                               !e         kT
                                                   dX

WT(X) is called the potential of mean force.




                  Potential of mean force

The potential of mean force can be re-written as:


                WT ( X ) = U P ( X ) + Wsol ( X )

Wsol(X) accounts implicitly and exactly for the effect of the solvent on the protein.

Implicit solvent models are designed to provide an accurate and fast

estimate of W(X).




                      Solvation Free Energy

                               Wsol
                  +                                      +

                                                            Sol
        Vac
    ! Wch                                                 Wch




                            Wnp


Wsol = Welec + Wnp = ( ch ! Wch )+ ( vdW + Wcav )
                     W sol    vac
                                   W




                                                                                        6
                         The SA model

Surface area potential


                                          N
                    Wcav + WvdW = ! " k SAk
                                       k =1




                         Eisenberg and McLachlan, (1986) Nature, 319, 199-203




                    Surface area potentials
                       Which surface?

                             Accessible
                                                                Molecular
                             surface
                                                                Surface




                  Hydrophobic potential:
                 Surface Area, or Volume?

                   Surface effect

                                               (Adapted from Lum, Chandler, Weeks,
                                               J. Phys. Chem. B, 1999, 103, 4570.)
                Volume effect



                            “Radius of the molecule”


  For proteins and other large bio-molecules, use surface




                                                                                     7
              Protein Electrostatics

• Elementary electrostatics
         • Electrostatics in vacuo
         • Uniform dielectric medium
         • Systems with boundaries


• The Poisson Boltzmann equation
         • Numerical solutions
         • Electrostatic free energies


• The Generalized Born model




    Elementary Electrostatics in vacuo


Some basic notations:


                          ! Fx ! Fy ! Fz                         Divergence
" • F = div F =  ()        !x
                              +
                                !y
                                   +
                                     !z
                  ( !f           !f   !f %                       Gradient
"f = grad ( f ) = &
                  ' !x           !y   !z #
                                         $
                                         !2 f !2 f !2 f          Laplacian
                 (             )
" • "f = div grad ( f ) = "2 f =             +    +
                                         !x 2 !y 2 !z 2




     Elementary Electrostatics in vacuo
Coulomb’s law:

The electric force acting on a point charge q2 as the result of the presence of
another charge q1 is given by Coulomb’s law:

                 r                                   q1q2
         q1          q2
                           u                 F=               u
                                                    4!" 0 r 2
Electric field due to a charge:

By definition:
                               F    q1                           q1
                      E=         =          u
                               q2 4!" 0 r 2

                                                              E “radiates”




                                                                                  8
      Elementary Electrostatics in vacuo
 Gauss’s law:

 The electric flux out of any closed surface is proportional to the
 total charge enclosed within the surface.
    Integral form:                            Differential form:

                     q                                             "(X )
      ! E • dA = "                             div ( E ( X )) =
                      0                                             !0

  Notes:
           - for a point charge q at position X0 , ρ(X)=q δ (X-X0)

           - Coulomb’s law for a charge can be retrieved from Gauss’s law




      Elementary Electrostatics in vacuo
Energy and potential:
    - The force derives from a potential energy U:


                             F = ! grad ( )
                                        U
    - By analogy, the electric field derives from an electrostatic potential φ:


                             E = " grad (! )
For two point charges in vacuo:                              Potential produced by q1 at
                                                             at a distance r:

                      q1q2                                                q1
        U=                                                     #=
                     4!" 0 r                                            4!" 0 r




      Elementary Electrostatics in vacuo
The cases of multiple charges: the superposition principle:

      Potentials, fields and energy are additive

 For n charges:                                                                      qN

                            N
                                          qi                       qi
             % (X ) = !
                            i =1    4#$ 0 X " X i
                                                                               X
                              N
                                               qi                                          q1
             E( X ) = !                                   u
                                                         2 i            q2
                             i =1   4#$ 0 X " X i
                                     qi q j
            U =!
                     i< j   4#$ 0 X " X i




                                                                                                9
        Elementary Electrostatics in vacuo

     Poisson equation:


                                    "
             div E =()              !0
                                                                             "
                    (                    )
             div grad (# ) = % • %(# ) = %2 (# ) = $
                                                                             !0

       Laplace equation:


                                    !2" = 0                     (charge density = 0)




              Uniform Dielectric Medium
Physical basis of dielectric screening
  An atom or molecule in an externally imposed electric field develops a non
  zero net dipole moment:
                                             -                    +

      (The magnitude of a dipole is a measure of charge separation)

     The field generated by these induced dipoles runs against the inducing
     field       the overall field is weakened (Screening effect)




                                                                                  The negative
                                                                                  charge is
                                                                                  screened by
                                                                                  a shell of positive
                                                                                  charges.




              Uniform Dielectric Medium
Electronic polarization:
                                     Under external               - -
                  - -   -                field                  -     -
              -             -                                 -
             -      +           -                           -         -
             -              -                               - +       -
              -                                               -
                  - -   -                                       - - -


                                                          Resulting dipole moment

Orientation polarization:


                                         Under external
                                             field




                                                             Resulting dipole moment




                                                                                                        10
                Uniform Dielectric Medium
Polarization:

       The dipole moment per unit volume is a vector field known as
       the polarization vector P(X).
                                                         " $1
       In many materials:         P( X ) = # E ( X ) =        E( X )
                                                          4!


  χ is the electric susceptibility, and ε is the electric permittivity, or dielectric constant

   The field from a uniform dipole density is -4πP, therefore the total field is


                               E = E applied # 4" P
                                      E applied
                               E=
                                         !




                Uniform Dielectric Medium

Some typical dielectric constants:
               Molecule                Dipole moment              Dielectric
                                        (Debyes) of a         constant ε of the
                                       single molecule         liquid at 20°C

               Water                          1.9                      80


               Ethanol                        1.7                      24


               Acetic acid                    1.7                      4


               Chloroform                    0.86                      4.8




                Uniform Dielectric Medium

  Modified Poisson equation:

                                                                 "
                         (             )
                  div grad (# ) = %2 (# ) = $
                                                                ! 0!
  Energies are scaled by the same factor. For two charges:


                                     q1q2
                         U=
                                    4"! 0!r




                                                                                                 11
              Uniform Dielectric Medium

The work of polarization:
   It takes work to shift electrons or orient dipoles.
   A single particle with charge q polarizes the dielectric medium; there is a
   reaction potential φ that is proportional to q for a linear response.

                               !R = Cq
   The work needed to charge the particle from q i =0 to q i =q:

              q                        q            1 2 1
    W = ! "R (qi )dqi = C ! qi dqi =                  Cq = q"R
             0                         0            2     2
   For N charges:
                                   1 N
                         W=          ! qi"iR               Free energy
                                   2 i =1




       System with dielectric boundaries

 The dielectric is no more uniform: ε varies, the Poisson equation becomes:


                                                                         " (X )
          (                            )
     div ! (X )grad (# ( X ) ) = % • ! (X )%# ( X ) = $
                                                                          !0


   If we can solve this equation, we have the potential, from which we can derive
   most electrostatics properties of the system (Electric field, energy, free energy…)

   BUT

   This equation is difficult to solve for a system like a macromolecule!!




         The Poisson Boltzmann Equation
ρ(X) is the density of charges. For a biological system, it includes the charges
of the “solute” (biomolecules), and the charges of free ions in the solvent:

                    ! ( X ) = ! solute ( X ) + ! ions ( X )
The ions distribute themselves in the solvent according to the electrostatic
potential (Debye-Huckel theory):
                         " qi! ( X )
       ni ( X )                            ni : number of ions of type i per unit volume
                =e           kT
                                           qi : charge on type i ion
          ni0
                   N
    " ions ( X ) = ! qi ni ( X )
                  i =1


  The potential f is itself influenced by the redistribution of ion charges, so the
  potential and concentrations must be solved for self consistency!




                                                                                           12
             The Poisson Boltzmann Equation
                                                                              q $( X )
                                                            & (X ) 1 N       " i
      # • % (X )#$ ( X ) = "                                      " ! qi ni0e kT
                                                             %0    % 0 i =1

  Linearized form:

                                                        $ (X )
          & • ! (X )&# ( X ) = %                               % ! ( X )" 2 ( X )# ( X )
                                                         !0

                                           N
                             1                                 2
          "2 =                             'n q   0 2
                                                  i i   =          I          I: ionic strength
                          ! 0!kT           i =1             ! 0!kT




    Solving the Poisson Boltzmann Equation

    • Analytical solution

         • Only available for a few special simplification of
           the molecular shape and charge distribution

    • Numerical Solution
         • Mesh generation -- Decompose the physical
           domain to small elements;
         • Approximate the solution with the potential value
           at the sampled mesh vertices -- Solve a linear
           system formed by numerical methods like finite
           difference and finite element method
         • Mesh size and quality determine the speed and
           accuracy of the approximation




                Linear Poisson Boltzmann equation:
                        Numerical solution

• Space discretized into a
  cubic lattice.

• Charges and potentials are
  defined on grid points.                                              εw
• Dielectric defined on grid lines

• Condition at each grid point:                                        εP
j : indices of the six direct neighbors of i

               6
                                     qi
             !# $
             j =1
                     ij     j   +
                                    # 0h
      $i =   6                                                  Solve as a large system of linear
             !#     ij
                                   2
                          + # ij" ij h 2                        equations
             j =1




                                                                                                    13
          Electrostatic solvation energy

The electrostatic solvation energy can be computed as an energy change
when solvent is added to the system:

                   1                 1
         Welec =     ! qi# RF (i ) = 2 ! qi (#S (i ) " #NS (i ))
                   2 i                 i


  The sum is over all nodes of the lattice

  S and NS imply potentials computed in the presence and absence of solvent.




       Approximate electrostatic solvation energy:
             The Generalized Born Model

                                   1 N
Remember:             "Gelec =       ! qi#iRF
                                   2 i =1
For a single ion of charge q and radius R:

                                 q2 & 1 #
                (GBorn =               $ ' 1!                           Born energy
                               8*) 0 R % )  "
For a “molecule” containing N charges, q1 ,…qN, embedded into spheres or radii
R1 , …, RN such that the separation between the charges is large compared to the
radii, the solvation energy can be approximated by the sum of the Born energy
and Coulomb energy:

                     N
                           qi2 ( 1 % 1 N N qi q j ( 1 %
        *Gelec = !                & ) 1# + !!                  & ) 1#
                    i =1 8,+ 0 Ri ' +  $ 2 i =1 j "i 4,+ 0 rij ' +  $




       Approximate electrostatic solvation energy:
             The Generalized Born Model

The GB theory is an effort to find an equation similar to the equation above,
that is a good approximation to the solution to the Poisson equation.
The most common model is:

                     1 (1 % N N                              qi q j
         )GGB =           & " 1#!!
                    8+* 0 ' *  $ i =1 j =1                              "
                                                                              rij2
                                                                            4 ai a j
                                                      rij2 + ai a j e

   ΔG GB is correct when rij      0 and r ij      ∞


  ai : Born radius of charge i:

  Assuming that the charge i produces a Coulomb potential:

                         1   1                    dV
                           =            !
                         ai 4"           r > Ri   r4




                                                                                       14
      Approximate electrostatic solvation energy:
            The Generalized Born Model

                                    &1 # 1
                                    $ ' 1!
                                    %(   " rij

               ΔG GB




                         Further reading
•   Michael Gilson. Introduction to continuum electrostatics.
    http://gilsonlab.umbi.umd.edu

•   M Schaefer, H van Vlijmen, M Karplus (1998) Adv. Prot. Chem., 51:1-57
    (electrostatics free energy)

•   B. Roux, T. Simonson (1999) Biophys. Chem., 1-278 (implicit solvent
    models)

•   D. Bashford, D Case (2000) Ann. Rev. Phys. Chem., 51:129-152
    (Generalized Born models)

•   K. Sharp, B. Honig (1990) Ann. Rev. Biophys. Biophys. Chem., 19:301-352
    (Electrostatics in molecule; PBE)

•   N. Baker (2004) Methods in Enzymology 383:94-118 (PBE)




         Energetics of protein structures

• Molecular Mechanics force fields

• Implicit solvent

• Statistical potentials




                                                                              15
                            Statistical Potentials


a

                            f(r)

    r

          b

                                                           r(Ǻ)




                                                  & P (r) #
                              E ( a, b, r ) = ' ln$ ( a ,b ) !
                                                  $          !
                                                  % P( r ) "
            Counts                                                        Energy
                  Ile-Asp                                                         Ile-Asp




                r(Ǻ)                                                       r(Ǻ)

                       Ile-Leu                                                    Ile-Leu




                r(Ǻ)                                                       r(Ǻ)




                    The Decoy Game
            Finding near native conformations
                                          1CTF
        Score




                                        cRMS (Ǻ)

                                                ' P( ai , a j , rij ) $
                       E = ! E (i, j ) = ( ! ln%                      "
                           i< j            i< j
                                                % P( r ) "
                                                &          ij         #




                                                                                            16