3D Equivariant Diffusion For Target-Aware Molecule Generation and Affinity Prediction

Targetdiff
ICLR 2023

1、Contributions

*一個(gè)端到端的框架，用于在蛋白靶點(diǎn)條件下生成分子，該框架明確考慮了蛋白質(zhì)和分子在三維空間中的物理相互作用。
*就我們所知，這是針對(duì)靶向藥物設(shè)計(jì)的第一個(gè)概率擴(kuò)散公式，其中訓(xùn)練和采樣過(guò)程以非自回歸和SE(3)-等變的方式對(duì)齊，這得益于移位中心操作和等變GNN。
*提出了幾個(gè)新的評(píng)估指標(biāo)和額外的見(jiàn)解，使我們能夠在許多不同的維度上評(píng)估模型生成的分子。實(shí)證結(jié)果證明了我們的模型優(yōu)于另外兩個(gè)代表性基準(zhǔn)模型。
*提出了一種基于我們的框架評(píng)估生成分子質(zhì)量的有效方法，其中模型可以作為評(píng)分函數(shù)來(lái)幫助排名，或者作為無(wú)監(jiān)督特征提取器來(lái)提高結(jié)合親和力預(yù)測(cè)的準(zhǔn)確性。

2、Problem definition

A protein binding site is represented as a set of atoms $P={(x_P^{(i)},v_P^{(i)})}_{i=1}^{N_P}$, where $N_P$ is the number of protein atoms, $x_P\in R^3$ represents the 3D coordinates of the atom, and $v_P\in R^{N_f}$ represents protein atom features such as element types and amino acid types. Our goal is to generate binding molecules $M={(x_L^{(i)},v_L^{(i)})}_{i=1}^{L_M}$ conditioned on the protein target. For brevity, we denote molecules as M = [x, v], where [·, ·] is the concatenation operator and $x\in R^{M\times 3}$ and $v\in R^{M\times K}$ denote atom Cartesian coordinates and one-hot atom types respectively.

3、Molecular diffusion process

use a Gaussian distribution $N$ to model continuous atom coordinates x and a categorical distribution C to model discrete atom types v. The atom types are constructed as a one-hot vector containing information such as element types and membership in an aromatic ring. We formulate the molecular distribution as a product of atom coordinate distribution and atom type distribution. At each time step t, a small Gaussian noise and a uniform noise across all categories are added to atom coordinates and atom types separately, according to a Markov chain with fixed variance schedules β1, . . . , βT (K為k維的平均噪聲向量)(實(shí)際上x(chóng)，v的調(diào)度不一致):

Denoting $\alpha t=1?{\beta }_t$ and
a desirable property of the diffusion process is to calculate the noisy data distribution $q(M_t|M_0)$ of any time step in closed-form(用閉合形式直接求出每個(gè)時(shí)間步時(shí)數(shù)據(jù)分布):
Using Bayes theorem, the normal posterior of atom coordinates and categorical posterior of atom types can both be computed in closed-form(通過(guò)貝葉斯公式求出后驗(yàn)分布):

4、Molecular generative process

The generative process, on reverse, will recover the ground truth molecule M0 from the initial noise MT , and we approximate the reverse distribution with a neural network parameterized by θ(t、P已知，Mt也已知，求μθ 、cθ):

There are different ways to parameterize ${\mu }_{\theta }([x_t,v_t],t,P)$ and $c_{\theta }([x_t,v_t],t,P)$. Here, we choose to let the neural network predict $[x_0,v_0]$ and feed it through equation 4 to obtain ${\mu }_{\theta }$ and $c_{\theta }$ which define the posterior distributions. we model the interaction between the ligand molecule atoms and the protein atoms with a SE(3)-Equivariant GNN:

At the l-th layer, the atom hidden embedding h(原子隱藏嵌入) and coordinates x(原子的坐標(biāo)) are updated alternately as follows:

where $d_{ij}=\Vert x_i?x_j\Vert$ is the euclidean distance(原子間歐幾里德距離) between two atoms i and j and eij is an additional feature(兩兩原子間連接特征，可以視為鄰接矩陣來(lái)描述原子之間的聯(lián)系或連接類型) indicating the connection is between protein atoms, ligand atoms or protein atom and ligand atom. 1mol is the ligand molecule mask since we do not want to update protein atom coordinates. The initial atom hidden embedding $h^0$ is obtained by an embedding layer that encodes the atom information. The final atom hidden embedding $h^L$ is fed into a multi-layer perceptron and a softmax function to obtain $?v_0$. Since $?x_0$ is rotation equivariant to $x_t$ and it is easy to see $x_{t?1}$ is rotation equivariant to $x_0$ according to equation 4, we achieve the desired equivariance for Markov transition.
注：the likelihood $p_{\theta }(M_0|P)$ should be invariant to translation and rotation of the protein-ligand complex. Denoting the SE(3)-transformation as $T_g$, we could achieve invariant likelihood w.r.t $T_g$ on the protein-ligand complex: $p_{\theta }(T_g(M_0|P))=p_{\theta }(M_0|P)$ if we shift the Center of Mass (CoM) of protein atoms to zero and parameterize the Markov transition $p(x_{t?1}|x_t,x_P)$ with an SE(3)-equivariant network.

5、Training

The combination of q and p is a variational auto-encoder (Kingma and Welling, 2013). The model can be trained by optimizing the variational bound on negative log likelihood. For the atom coordinate loss, since $q(x_{t?1}|x_t,x_0)$ and $p_{\theta }(x_{t?1}|x_t)$ are both Gaussian distributions, the KL-divergence can be written in closed form:

where
and $C$ is a constant. In practice, training the model with an unweighted MSE loss (set ${\gamma }_t$ = 1) could also achieve better performance as Ho et al. (2020) suggested. For the atom type loss, we can directly compute KL-divergence of categorical distributions as follows:

The final loss is a weighted sum of atom coordinate loss and atom type loss: $L=L_{t?1}^{(x)}+\lambda L_{t?1}^{(v)}$. We summarize the overall training and sampling procedure of TargetDiff in Appendix E.
(1) training

(2) sampling
At the l-th layer, we dynamically construct the protein-ligand complex as a k-nearest neighbors (knn) graph based on known protein atom coordinates and current ligand atom coordinates, which is the output of the l ? 1-th layer. We choose k = 32 in our experiments. The protein atom features include chemical elements, amino acid types and whether the atoms are backbone atoms. The ligand atom types are one-hot vectors consisting of the chemical element types and aromatic information. The edge features are the outer products of distance embedding and bond types, where we expand the distance with radial basis functions located at 20 centers between 0 ? A and 10 ? A and the bond type is a 4-dim one-hot vector indicating the connection is between protein atoms, ligand atoms, protein-ligand atoms or ligand-protein atoms.

6、Experiments

Data：Crossocked2022
Baseline：liGAN、AR、Pocket2Mol、GraphBP
Targetiff：Our model contains 9 equivariant layers described in equation 7, where fh and fx are specifically implemented as graph attention layers with 16 attention heads and 128 hidden features. We first decide on the number of atoms for sampling by drawing a prior distribution estimated from training complexes with similar binding pocket sizes. After the model finishes the generative process, we then use OpenBabel (O’Boyle et al., 2011) to construct the molecule from individual atom coordinates as done in AR and liGAN.

7、Results

8、Target Binding Affinity

We first establish the connection between unsupervised generative models and binding affinity ranking / prediction. Under our parameterization, the network predicts the denoised $[?x_0,?v_0]$. Given the protein-ligand complex, we can feed ${\phi }_{\theta }$ with $[x_0,v_0]$ while freezing the x-update branch (i.e. only atom hidden embedding $h$ is updated), and we could finally obtain $h^L$ and $?v_0$:

Our assumption is that if the ligand molecule has a good binding affinity to protein, the flexibility of atom types should be low, which could be reflected in the entropy of $?v_0$(v_ent). Therefore, it can be used as a scoring function to help ranking, whose effectiveness is justified in the experiments. In addition, hL also includes useful global information. We found the binding affinity ranking performance can be greatly improved by utilizing this feature with a simple linear transformation.
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