Thank you for the review! To address your questions and concerns (our updated manuscript can be downloaded at the top of this page):

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Regarding introduced metric: "other algorithms under this metric, or stronger domain justification"

We assume that you refer to our compositionality metric and are glad you find it intuitive! We added Table 4 to the revised manuscript, where we evaluate additional protein structure generative models' compositionality. We note that an ellipsoid conditioned model could be built out of any protein structure generative model, and the base model does not necessarily have to be Multiflow.

Further, we add an explanation to Appendix B: that our compositionality metric is inspired by the ``Diversity Index" (https://en.wikipedia.org/wiki/Diversity_index), which is often used in the domains of ecology or demography.

Formatting and white space in the appendix

Thank you for pointing it out! We fixed it in our revision.

User guidelines for choosing the number of ellipsoids $K$

That is a nice suggestion! We added Figure 11 with a histogram of $K$ for PDB proteins and this guidance to Section B: All specifications of numbers of ellipsoids $K<10$ can be expected to yield successful generations. To see which $K$ are the most in-distribution, please see the histogram in Figure 11 where we visualize the frequency of different $K$.

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We hope the discussions and results address your concerns! Please let us know if there are any further opportunities to improve the score.

Thank you for the review! To address your questions and concerns (our updated manuscript can be downloaded at the top of this page):

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On the applicability of the ellipsoid conditioning on practical protein design tasks

An important question! We added the following to Appendix B:

• Example use case: We aim to scaffold a therapeutically relevant functional site. The protein requires a certain shape to fit into a delivery mechanism. With ProtComposer we can specify the rough shape and size of the scaffold to still fit into the delivery mechanism
• ProtComposer can redesign the connectivity of secondary structure elements: biologists aim to escape the existing space of protein topologies and discover new ones that can be used as scaffolds or for other design tasks.
• Example use case: We aim to design a binder for a target at a flat beta-sheet region. With ProtComposer, we can specify that a beta-sheet of the right size and shape should interface with the target's beta-sheet to increase the probability of success in generating a strong binder.
• We often know how much flexibility/rigidity we want in certain areas of the protein. With ProtComposer, we can place a rigid helix bundle, a beta-barrel, or more loosely connected substructures in those regions.

How is the length between ellipsoids determined/sampled?

We add the following clarification to A.1:

At inference time, ProtComposer (and Multiflow, RFDiffusion, and Chroma) take a protein length $L$ as input which specifies the number of residues in the generated protein. In ProtComposer, each ellipsoid $\mathbf{E}_k$ is associated with a number of residues $n_k$ that is supposed to end up in that ellipsoid. The sum of $n_k$ does not have to match the total lenght $L$. The $n_k$ are just an additional conditioning input which the model can adhere to but does not have to. Even if the sum of $n_k$ is larger than $L$, the model can still (and does) use some of the residues for strands and to connect the ellipsoids. When generating from synthetic ellipsoids, we set $L$ to be equal to the sum of $n_k$. When generating from data-extracted ellipsoids, we set $L$ to be the length of the original protein which the ellipsoids were extracted from, so $L$ > $\sum n_k$.

"provide ablation study results on the effect of self-conditioning, particularly the two self-conditioning schemes described in line 290."

We added Section C.2 with Figure 9 to the revised manuscript which shows the designability vs. ellipsoid adherence frontier under our 5 different possibilities of performing self-conditioning, showing that the interpolation variant performs best under all settings of guidance strength.

Figure 9: What is the linear fit and statistical significance in both cases?

We added to the Figure's caption that the linear fit for alpha-helices is 0.97 and for beta-sheets 0.93 and that for both, the statistical significance is high with a p-value that is so close to 0 as to be outside of numerical precision.

"During training, is the ellipsoid conditioning information always provided, or only provided for a percentage of time?"

For classifier free guidance we combine a conditional and an unconditional model. During pretraining, to obtain the unconditional Multiflow, there is never any ellipsoid conditioning. When fine-tuning Multiflow to obtain the conditional model, ellipsoid conditioning is always provided.

Why is a structured residue considered as “covered” if it is inside at least one ellipsoid instead of inside the ellipsoid it is assigned?

The notion of a residue being assigned to an ellipsoid does not exist at inference time. The ellipsoids only have an assigned number of residues, no assignment of which residue will go into them. The model can freely choose which residue ends up in which ellipsoid.

Intuition on why "over-guidance" with $\lambda > 1$ can yield improvements (Table 1)

A very interesting question! This is a long-standing and not fully understood phenomenon of diffusion/flow models. Figure 2 in the classifier-free guidance paper (https://arxiv.org/pdf/2207.12598) provides intuition by showing what happens for a Gaussian mixture - with "over-guidance" we make "extra-sure" that the samples are close to the conditioning distribution, and far from other possibilities. Most justification is empirical and, e.g., this blog post's Section 3 (https://sander.ai/2022/05/26/guidance.html) shows how image quality is improved with "over-guidance" where $\lambda = 3$. Lastly, the autoguidance paper (https://arxiv.org/pdf/2406.02507v1) observes that improvements are even possible when guiding a model with a worse, less trained version of itself.

Whether our "PDB protein" evaluation set is included in the training data

This set of proteins is separate from the training data.

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We hope the discussions and results address your concerns! Please let us know if there are any further opportunities to improve the score.

Thank you for the review! To address your questions and concerns (our updated manuscript can be downloaded at the top of this page):

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Sequence-structure co-generation results

We added Table 3 where we select statistical model parameters with high designability ($\nu = 50, \sigma=5$) and draw 400 structures together with sequences from ProtComposer. As in the Multiflow paper, the joint generation does not improve designability. Interestingly, the median self-consistency RMDSs (scRMSD) of the jointly generated sequences are worse, while their mean scRMSDs are better.

1-seq Designability refers to generating 1 sequence per structure, while 8-seq Designability uses the best out of 8 sequences per structure. Self-consistency RMSD is abbreviated as scRMSD.

Ellipsoid segmentation cutoff

We added Figure 14 to the paper, which visualizes the effect of different radius cutoffs on ellipsoids. We observe that a 5A cutoff offers reasonable 3D ellipsoid segmentations that are neither too coarse nor too fine-grained.

Communication between ellipsoid and residue tokens

We added this discussion to Appendix B:

In our transformer layers (algorithm 2), the residue tokens update all ellipsoid tokens, and ellipsoid tokens update all residue tokens. In Invariant-Cross-Attention (algorithm 1), we inject ellipsoid position and geometry information into the residues tokens without updating ellipsoid tokens.

The reason: ICA updates a residue token based on transforming the ellipsoid means and covarianc matrices into the local coordinate frame of the residue (where residue frames are defined as in AlphaFold2). The same mechanism is not applicable to update ellipsoid tokens based on residue positions since a canonical assignment of a frame to an ellipsoid is not possible. To see this, we consider the worst-case scenario of a round ellipsoid - clearly no canonical assignment of a 3D orientation is possible. In the best-case scenario of an ellipsoid with three distinctly sized principal components, we could choose, e.g., the two largest to construct a frame from. However, their sign is arbitrary, leading to 4 options among which no canonical choice exists. Thus, we opted for our ICA without ellipsoid token updates, which is empirically sufficient for strong ellipsoid adherence and, together with our transformer layers, aligns with the mechanisms commonly employed in computer vision [1,2,3].

"Is it possible to set the order of ellipsoids?"

A simple approach to allow for setting the order would be retraining a model with an additional ellipsoid feature that encodes a list of the orders in which the ellipsoid is "hit".

Further ablation results

Since you advised adding ablations as a last avenue to improve the paper further, we also added an investigation of different self-conditioning approaches. We added Section C.2 with Figure 9 to the revised manuscript which shows the designability vs. ellipsoid adherence frontier under our 5 different possibilities of performing self-conditioning.

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We hope the discussions and results address your concerns! Please let us know if there are any further opportunities to improve the score.

[1] Adding Conditional Control to Text-to-Image Diffusion Models
[2] GLIGEN: Open-Set Grounded Text-to-Image Generation
[3] Compositional Text-to-Image Generation with Dense Blob Representations

3: Results in table 2, the impact of guidance on the tradeoff between designability vs. helicity+diversity+novelty, and comparisons to RFDiffusion, Chroma, and Multiflow.

We removed our mention of "natural proteins," which may have been confusing and could leave the impression that there is a notion of comparing models on synthetic vs. natural proteins, which is not the case. All models (ProtComposer, RFDiffusion, Chroma, Multiflow) can sample a distribution of protein structures p(X) without any further inputs. In ProtComposer, this distribution is decomposed as p(X)=P(X|E)p(E) into a distribution of ellipsoids p(E) and a distribution of structures conditioned on ellipsoids P(X|E).

In Figure 4 we generate proteins from scratch with all methods. Table 2 shows numbers for ProtComposer when generating from a special type of ellipsoids - ellipsoids extracted from PDB proteins. This would not be available when attempting to generate a protein from scratch.

Regarding the guidance and the tradeoff: We introduce the guidance mechanism as a control knob for trading off between designability vs. helicity, diversity, and novelty. In other models, such as RFDiffusion, this tradeoff can be controlled via the sampling temperature. We compare the strength of the tradeoffs that can be achieved for all models via their tradeoff curves in Figure 4. It shows ProtComposer's improved tradeoff between designability and the other metrics, including helicity if lower helicity is the goal (note the minus sign in "1 - helicity" on the y-axis).

4: Self-conditioning via interpolation vs. other variants

We added Section C.2 with Figure 9 to the revised manuscript which shows the designability vs. ellipsoid adherence frontier under our 5 different possibilities of performing self-conditioning, showing that the interpolation variant performs best under all settings of guidance strength.

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We hope the discussions and additions address your concerns! When appropriate, we also reference our additions and clarifications in the main text. Please let us know if there are any further opportunities to improve the score.

[1] Adding Conditional Control to Text-to-Image Diffusion Models
[2] GLIGEN: Open-Set Grounded Text-to-Image Generation
[3] Compositional Text-to-Image Generation with Dense Blob Representations
[4] Self-Attention with Relative Position Representations

Thank you for the review! To address your questions and concerns (our updated manuscript can be downloaded at the top of this page):

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Limitation to secondary structure conditioning

We acknowledge that conditioning on ellipsoids with functional annotations beyond secondary structure would be highly interesting. We considered the feasibility of this direction during the project using data from function annotation databases such as Interpro or using predicted annotations from models such as DeepFRI. However, the data quality is insufficient for training a model conditioned on ellipsoids with function annotations. For instance, in Interpro, many location-specific annotations are assigned to the whole protein instead of the functionally relevant site or the DeepFRI location predictions conflict with each other for different functions. Thus, we provide a general framework that allows for arbitrary annotation types but only implement it for secondary structure annotations - once sufficient quality data of broader annotations is available, the more general model can be trained.

1.a.1: "Is k determined by the structure of the training protein? If so, what is the distribution of k in natural proteins?"

We now clarify in A.1 that during training, the data determines the number of ellipsoids and added Figure 11 with a histogram of $K$.

1.a.2: $K=5$ for the synthetic ellipsoid experiments

We added a clarification in A.4 that we chose a fixed $K$ for consistency across protein lengths and the specific value of $K=5$ since it is the most frequent number of ellipsoids in PDB proteins (see Figure 11). We added Figure 17 to the paper where we show proteins generated with different $K$.

1.b: "The current representation specifies the number of residues in each ellipsoid, but the authors show that this number directly depends on ellipsoid volume. Could specifying the number of residues be redundant, and might removing this constraint provide the model more flexibility in generation?"

We added to the discussion in Appendix B that this is a reasonable alternative and that we chose to specify the number of residues per ellipsoid to give the user the option of controlling it (or having the number be determined via linear fit if preferred).

2.a: The authors separately model the SE3 features (E_k) and scalar features (e_k) of ellipsoids using the proposed ICA and transformer to achieve SE3 invariance in the local frame. Has the team considered alternative approaches, such as modeling ellipsoids as “pseudo frames” with SE3 and scalar features and simply using IPA to update ellipsoid and residue features together?

We added this discussion to Appendix B:

We indeed considered this. However, a canonical assignment of a frame to an ellipsoid is not possible. To see this, we consider the worst-case scenario of a round ellipsoid - clearly, no canonical assignment of a 3D orientation is possible. In the best-case scenario of an ellipsoid with three distinctly sized principal components, we could choose, e.g., the two largest to construct a frame from. However, their sign is arbitrary, leading to 4 options among which no canonical choice exists. Thus, we opted for our ICA without ellipsoid token updates, which is empirically sufficient for strong ellipsoid adherence and, together with our transformer layers, aligns with the mechanisms commonly employed in computer vision [1,2,3].

2.b.1: clarifing PosEmbed in Alg. 1

We added to A.1 in the revised manuscript that PosEmbedd is a sinusoidal positional encoding of the relative positions (the vectors between ellipsoid means and residue positions). Each number of the 3D offset vector is encoded into 64 dimensions, and all 3 are concatenated.

2.b.2: On why the query uses un-updated s, while the key and value use a, which incorporates relative positional information

We added to Appendix B that this approach to inject relative positional encoding was our default choice since it is how relative positional encodings are used in language model transformers [4] or in geometric transformers such as "SE3-Transformer".