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Equation 4 · How AI for Science and Medicine Actually Works

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xt=αˉt x0+1−αˉt ϵ,x^0=fθ(xt,t),x_t = \sqrt{\bar{\alpha}_t}\, x_0 + \sqrt{1-\bar{\alpha}_t}\, \epsilon, \qquad \hat{x}_0 = f_\theta(x_t, t),

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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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xtx_t

Symbol x_t

a coordinate in the stated process.

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αˉt\bar{\alpha}_t

Symbol barα_t

barα_t is one of the signed contributions combined to compute the quantity on the left.

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x0x_0

Symbol x_0

a coordinate in the stated process.

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ϵ\epsilon

Symbol epsilon

a random quantity drawn from a standard normal distribution.

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x^0\hat{x}_0

Symbol hatx_0

hatx0x_0 is one of the signed contributions combined to compute the quantity on the left.

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fθf_\theta

Symbol f_θ

the trained network.

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tt

Symbol t

the noise-level index.

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=

=

The expressions on both sides represent the same quantity under the stated assumptions.

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√

√

Take a square root.

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addition

addition

Add the term after the plus sign to the term or group before it.

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subscript

subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

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What the article says around this equation

The larger change is in how coordinates are produced. Rather than a Structure Module built on rotations and translations, AlphaFold3 uses a diffusion module that operates directly on raw atom coordinates and a coarse token representation, without rotational frames at all. Diffusion models are trained on a simple, general task: take a true data point, corrupt it with noise at a known level, and train a network to recover the original from the corrupted version. For coordinates x0x_0 noised to xtx_t at noise level t , xt=αˉt x0+1−αˉt ϵ,x^0=fθ(xt,t)x_t = \sqrt{\bar{\alpha}_t}\, x_0 + \sqrt{1-\bar{\alpha}_t}\, \epsilon, \qquad \hat{x}_0 = f_\theta(x_t, t). with ϵ\epsilon drawn from a standard normal distribution and fθf_\theta the trained network. Generating a structure then means starting from coordinates that are…
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The larger change is in how coordinates are produced. Rather than a Structure Module built on rotations and translations, AlphaFold3 uses a diffusion module that operates directly on raw atom coordinates and a coarse token representation, without rotational frames at all. Diffusion models are trained on a simple, general task: take a true data point, corrupt it with noise at a known level, and train a network to recover the original from the corrupted version. For coordinates x0x_0 noised to xtx_t at noise level t , xt=αˉt x0+1−αˉt ϵ,x^0=fθ(xt,t)x_t = \sqrt{\bar{\alpha}_t}\, x_0 + \sqrt{1-\bar{\alpha}_t}\, \epsilon, \qquad \hat{x}_0 = f_\theta(x_t, t). with ϵ\epsilon drawn from a standard normal distribution and fθf_\theta the trained network. Generating a structure then means starting from coordinates that are almost pure noise and repeatedly applying the trained denoiser, each pass nudging the atoms closer to a physically coherent arrangement, across many scales at once — from the local geometry of a single ring to the global fold of an entire complex [ 2 ] . Both AlphaFold2 and AlphaFold3 were trained on structures deposited in the Protein Data Bank; AlphaFold3’s training data was cut at structures released before 30 September 2021, supplemented by sequence databases including UniRef90, Uniclust30, and BFD, and RNA-specific databases including Rfam and RNAcentral [ 2 ] .

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