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Published equation contexts

p(x)=e−βF(x)∫e−βF(x′) dx′,β=1kBTp(x) = \frac{e^{-\beta F(x)}}{\displaystyle\int e^{-\beta F(x')}\,dx'}, \qquad \beta = \frac{1}{k_{\mathrm B}T}

Why this formula appears here

A protein in solution is not a coordinate file. It is a distribution. Under equilibrium conditions the physically meaningful object is a Boltzmann-weighted ensemble over conformations, p(x)=e−βF(x)∫e−βF(x′) dx′,β=1kBTp(x) = \frac{e^{-\beta F(x)}}{\displaystyle\int e^{-\beta F(x')}\,dx'}, \qquad \beta = \frac{1}{k_{\mathrm B}T}. and function frequently depends on the populations and interconversion rates of its modes rather than on the single most populated one. A predicted structure is a point estimate. It carries no populations, no relative free energies, and no exchange kinetics — none of which can be recovered from the file by inspection.

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pp

Symbol p

p occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

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e−βF(x)e^{-\beta F(x)}

Symbol e^-β F(x)

e−e^-β F(x) occurs above the fraction bar. The numerator is divided by the entire denominator below it.

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e−βF(x′)e^{-\beta F(x')}

Symbol e^-β F(x')

e−e^-β F(x') occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

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dd

Symbol d

d occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

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kBk_{\mathrm B}

Symbol k_mathrm B

kmk_mathrm B occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

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TT

Symbol T

T occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

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∫e−βF(x′) dx′\displaystyle\int e^{-\beta F(x')}\,dx'

Denominator: displaystyleint e^-β F(x')dx'

The complete quantity below the fraction bar; it must be nonzero for this division.

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kBTk_{\mathrm B}T

Denominator: k_mathrm BT

The complete quantity below the fraction bar; it must be nonzero for this division.

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How to interpret it

The result depends on every term or point included by the summation or integral. With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

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Published contexts (1)

A symbol can carry a different meaning in another article. Each occurrence keeps its own guide and term definitions.

p(x)=e−βF(x)∫e−βF(x′) dx′,β=1kBT,p(x) = \frac{e^{-\beta F(x)}}{\displaystyle\int e^{-\beta F(x')}\,dx'}, \qquad \beta = \frac{1}{k_{\mathrm B}T},

Equation 12 · Structural Biology

Structure from Sequence: What Protein Folding Prediction Did and Did Not Settle

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

A protein in solution is not a coordinate file. It is a distribution. Under equilibrium conditions the physically meaningful object is a Boltzmann-weighted ensemble over conformations, p(x)=e−βF(x)∫e−βF(x′) dx′,β=1kBTp(x) = \frac{e^{-\beta F(x)}}{\displaystyle\int e^{-\beta F(x')}\,dx'}, \qquad \beta = \frac{1}{k_{\mathrm B}T}. and function frequently depends on the populations and interconversion rates of its modes rather than on the single most populated one. A predicted structure is a point estimate. It carries no populations, no relative free energies, and no exchange kinetics — none of which can be recovered from the file by inspection.

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