Leapfrog: how to solve Newton’s 2 nd Law on the computer

Leapfrog: how to solve Newton’s 2nd Law on the computer

credit: Zhijun Wu, Department of Mathematics, Iowa State University

Newton’s equations

nixxfdt

xdm ni

ii ,...,1),,...,( 12

2

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f i(x1,...,xn ) = −∂U(x1,...,xn ) /∂x i, i =1,...,n

nifam iii ,...,1,

Leapfrog: how to solve Newton’s 2nd Law on the computercredit: Zhijun Wu, Department of Mathematics, Iowa State University

Newton’s equations: reduce to a pair of first order equations

nivdt

dx

xxfdt

dvm

ii

nii

i

,...,1,

),,...,( 1


Newton’s equations: numerical solution

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miv ik+1 − v i

k

t k+1 − t k≈ f i(x1

k,...xnk )

x ik+1 − x i

k

t k+1 − t k≈ v i

k, i =1,...,n

t

x

tk tk+1

xk

xk+1

tk tk+1

vk vk+1

t

vx(t) v(t)

kikk

ki

ki v

tt

xx

1

1kikk

ki

ki a

tt

vv

1

1


Newton’s equations: leapfrog algorithm

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for k =1,...,m

v ik+ 12 = v i

k− 12 +f i(x1

k,...xnk )

miΔt

x ik+1 = x i

k + v ik+ 12Δt, i =1,...,n

end

ttttttttt kkkkkk 21

211 2

121

,,

xk-1 xk xk+1

vk-1/2 vk+1/2

The force field: what is f? (which comes from U, so what is U?)Credit: Alexander D. MacKerell, Jr. University of Maryland, Baltimore School of Pharmacy

The energy of chemical structures, in combination with statistical mechanics, allows for, in principle, all properties of a system to be calculated. In practice, however, this is limited by the inability to calculate the energy of all possible conformations of a chemical system. One way to overcome this limitation is the use of simple mathematical functions to treat the structure-energy relationship; this type of approach is referred to as molecular mechanics or empirical force field calculations.

However, the equation alone does not allow for computation of structure-energy relationships. In addition, parameters must be included in the mathematical equation. Different parameters allow for the same mathematical equation to be applied to different chemical entities. Further, the “quality” of these parameters dictate the validity of the computed structure-energy relationships.

The force fieldCredit: Alexander D. MacKerell, Jr. University of Maryland, Baltimore School of Pharmacy

The energy of chemical structures, in combination with statistical mechanics, allows for, in principle, all properties of a system to be calculated. A FORCE FIELD uses simple mathematical functions to treat the structure-energy relationship

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E torsion



Empirical Force Field Parameter SetsExample: CHARMM -- Proteins, nucleic acids, lipids, carbohydrates and a variety of small model compounds, including coenzymes


Empirical Force Field Parameter SetsExample: CHARMM -- Proteins, nucleic acids, lipids, carbohydrates and a variety of small model compounds, including coenzymes

The use of Coulomb’s law with fixed atomic charges to treat the electrostatic interactions is a major simplification in current force fields.

It is well known that the electron distribution of a molecule (and, thus, the atomic charges) changes as a function of the electrostatic field around the molecule. This is ignored in additive force fields.

To compensate for this omission, the atomic charges are “enhanced” to mimic the polarization of molecules that occurs in a polar, condensed phase environment (e.g. aqueous solution, TIP3P water model dipole moment = 2.35 versus gas phase value of 1.85).

Limitation of additive force fields

The nonbond/intermolecular parameters will impact the resulting geometries, vibrations and conformational energies. Thus, it is necessary to apply an iterative approach


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Leapfrog: how to solve Newton’s 2 nd Law on the computer