CyclicCAE: A Conformational Autoencoder for Efficient Heterochiral Macrocyclic Backbone Sampling

CyclicCAE: A Conformational Autoencoder for Efficient Heterochiral Macrocyclic Backbone Sampling | bioRxiv /* */ /* */ <!-- <!-- /*! * yepnope1.5.4 * (c) WTFPL, GPLv2 */ (function(a,b,c){function d(a){return"[object Function]"==o.call(a)}function e(a){return"string"==typeof a}function f(){}function g(a){return!a||"loaded"==a||"complete"==a||"uninitialized"==a}function h(){var a=p.shift();q=1,a?a.t?m(function(){("c"==a.t?B.injectCss:B.injectJs)(a.s,0,a.a,a.x,a.e,1)},0):(a(),h()):q=0}function i(a,c,d,e,f,i,j){function k(b){if(!o&&g(l.readyState)&&(u.r=o=1,!q&&h(),l.onload=l.onreadystatechange=null,b)){"img"!=a&&m(function(){t.removeChild(l)},50);for(var d in y[c])y[c].hasOwnProperty(d)&&y[c][d].onload()}}var j=j||B.errorTimeout,l=b.createElement(a),o=0,r=0,u={t:d,s:c,e:f,a:i,x:j};1===y[c]&&(r=1,y[c]=[]),"object"==a?l.data=c:(l.src=c,l.type=a),l.width=l.height="0",l.onerror=l.onload=l.onreadystatechange=function(){k.call(this,r)},p.splice(e,0,u),"img"!=a&&(r||2===y[c]?(t.insertBefore(l,s?null:n),m(k,j)):y[c].push(l))}function j(a,b,c,d,f){return q=0,b=b||"j",e(a)?i("c"==b?v:u,a,b,this.i++,c,d,f):(p.splice(this.i++,0,a),1==p.length&&h()),this}function k(){var a=B;return a.loader={load:j,i:0},a}var l=b.documentElement,m=a.setTimeout,n=b.getElementsByTagName("script")[0],o={}.toString,p=[],q=0,r="MozAppearance"in l.style,s=r&&!!b.createRange().compareNode,t=s?l:n.parentNode,l=a.opera&&"[object Opera]"==o.call(a.opera),l=!!b.attachEvent&&!l,u=r?"object":l?"script":"img",v=l?"script":u,w=Array.isArray||function(a){return"[object Array]"==o.call(a)},x=[],y={},z={timeout:function(a,b){return b.length&&(a.timeout=b[0]),a}},A,B;B=function(a){function b(a){var a=a.split("!"),b=x.length,c=a.pop(),d=a.length,c={url:c,origUrl:c,prefixes:a},e,f,g;for(f=0;f<d;f++)g=a[f].split("="),(e=z[g.shift()])&&(c=e(c,g));for(f=0;f<b;f++)c=x[f](c);return c}function g(a,e,f,g,h){var i=b(a),j=i.autoCallback;i.url.split(".").pop().split("?").shift(),i.bypass||(e&&(e=d(e)?e:e[a]||e[g]||e[a.split("/").pop().split("?")[0]]),i.instead?i.instead(a,e,f,g,h):(y[i.url]?i.noexec=!0:y[i.url]=1,f.load(i.url,i.forceCSS||!i.forceJS&&"css"==i.url.split(".").pop().split("?").shift()?"c":c,i.noexec,i.attrs,i.timeout),(d(e)||d(j))&&f.load(function(){k(),e&&e(i.origUrl,h,g),j&&j(i.origUrl,h,g),y[i.url]=2})))}function h(a,b){function c(a,c){if(a){if(e(a))c||(j=function(){var a=[].slice.call(arguments);k.apply(this,a),l()}),g(a,j,b,0,h);else if(Object(a)===a)for(n in m=function(){var b=0,c;for(c in a)a.hasOwnProperty(c)&&b++;return b}(),a)a.hasOwnProperty(n)&&(!c&&!--m&&(d(j)?j=function(){var a=[].slice.call(arguments);k.apply(this,a),l()}:j[n]=function(a){return function(){var b=[].slice.call(arguments);a&&a.apply(this,b),l()}}(k[n])),g(a[n],j,b,n,h))}else!c&&l()}var h=!!a.test,i=a.load||a.both,j=a.callback||f,k=j,l=a.complete||f,m,n;c(h?a.yep:a.nope,!!i),i&&c(i)}var i,j,l=this.yepnope.loader;if(e(a))g(a,0,l,0);else if(w(a))for(i=0;i (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0];var j=d.createElement(s);var dl=l!='dataLayer'?'&l='+l:'';j.src='//www.googletagmanager.com/gtm.js?id='+i+dl;j.type='text/javascript';j.async=true;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-M677548'); Skip to main content Home About Submit ALERTS / RSS Search for this keyword Advanced Search New Results CyclicCAE: A Conformational Autoencoder for Efficient Heterochiral Macrocyclic Backbone Sampling View ORCID Profile Andrew C. Powers , View ORCID Profile P. Douglas Renfrew , View ORCID Profile Parisa Hosseinzadeh , View ORCID Profile Vikram Khipple Mulligan doi: https://doi.org/10.1101/2025.02.21.639569 Andrew C. Powers 1 Department of Bioengineering University of Oregon , Eugene, Oregon Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Andrew C. Powers P. Douglas Renfrew 2 Center for Computational Biology Flatiron Institute , New York, New York Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for P. Douglas Renfrew Parisa Hosseinzadeh 1 Department of Bioengineering University of Oregon , Eugene, Oregon Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Parisa Hosseinzadeh For correspondence: parisah{at}uoregon.edu vmulligan{at}flatironinstitute.org Vikram Khipple Mulligan 2 Center for Computational Biology Flatiron Institute , New York, New York Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Vikram Khipple Mulligan For correspondence: parisah{at}uoregon.edu vmulligan{at}flatironinstitute.org Abstract Full Text Info/History Metrics Preview PDF Abstract Macrocycles are a promising therapeutic class. The incorporation of heterochiral and non-natural chemical building-blocks presents challenges for rational design, however. With no existing machine learning methods tailored for heterochiral macrocycle design, we developed a novel convolutional autoencoder model to rapidly generate energetically favorable macrocycle backbones for heterochiral design and structure prediction. Our approach surpasses the current state-of-the-art method, Generalized Kinematic loop closure (GenKIC) in the Rosetta software suite. Given the absence of large, available macrocycle datasets, we created a custom dataset in-house and in silico . Our model, CyclicCAE, produces energetically stable backbones and designable structures more rapidly than GenKIC. It enables users to perform energy minimization, generate structurally similar or diverse inputs via MCMC, and conduct inpainting with fixed anchors or motifs. We propose that this novel method will accelerate the development of stable macrocycles, speeding up macrocycle drug design pipelines. 1 Introduction Cyclic peptide macrocycles are a promising class of therapeutics due to their distinct advantages over linear peptides, including enhanced conformational stability, greater protection against proteolytic degradation [ 1 , 2 ], and improved membrane permeability [ 2 , 3 , 4 , 5 , 6 , 7 ]. Solid-phase peptide synthesis enables the incorporation of not only the 20 canonical L-α-amino acids, but of non-canonical amino acid (NCAA) side chains and backbones, which can enhance cell permeability, provide structural rigidity, or confer other desired properties. The use of both L- and D-amino acids in heterochiral macrocycles provides resistance to proteolytic cleavage [ 3 ], and opens a broader conformational landscape with richer structural diversity, increasing the chances of finding new, functional peptide folds for a given application. Other non-canonical building blocks, such as N -methylated amino acids or peptoid monomers, serve to remove backbone hydrogen bond donor groups, enhancing membrane permeability — an important property for macrocycle drugs [ 8 ]. Macrocycles have not benefited as much from rapid advancements in machine learning (ML) as proteins have[ 9 , 10 , 11 , 12 ], primarily due to the lack of training data available within the Protein Data Bank (PDB). Current approaches that adapt existing models for macrocycles, such as AfCycDesign [ 13 ], RFPeptide [ 14 ], and HighFold [ 15 ] have however shown success, but only for a subset of building-block types. For example, AfCycDesign predicts the structure of L-α-homochiral canonical macrocycles by modifying the positional encoding of AlphaFold2 with a cyclic offset. This allowed the authors to generate L-α-homochiral macrocycles by hallucinating through different sequences and minimizing the distogram predicted by AlphaFold2 to the desired structure [ 13 ]. All three methods rely on initial weights trained exclusively on canonical proteins from the PDB, and consequently cannot predict or design structures of heterochiral macrocycles, for which large databases of experimental training data are not available. To date, the authors are unaware of any machine learning approaches that have been developed to generate, predict structures of, or assign sequences to heterochiral macrocycles. Physics-based methods that perform semi-random sampling in a manner that is less biased by prior knowledge, such as the peptide macrocycle workflows implemented in the Rosetta software suite, are currently the state of the art for designing and predicting structures of heterochiral and noncanonical macrocycles (reviewed in [ 16 , 17 , 18 ]). These have been successfully applied to create a wide range of well-folded macrocycles [ 19 , 20 , 21 , 22 , 23 ], including macrocycles able to bind to and inhibit targets of therapeutic relevance, such as the New Delhi metallo- β -lactamase [ 24 ] or histone deacetylases [ 25 ]. Within the existing Rosetta macrocycle design pipeline, the current method for macrocycle backbone conformational sampling is Rosetta’s Generalized Kinematic Closure (GenKIC) algorithm [ 19 , 26 ]. Kinematic closure, an algorithm borrowed from the robotics field, solves analytically for the internal degrees of freedom of a chain of atoms in order to connect the start and end of a loop or a closed cycle [ 27 , 28 ]. GenKIC segments a loop by designating three atoms as pivot points, and allowing the remaining residues to sample torsion angles guided by the Ramachandran preferences of each amino acid. Once segment torsions have been sampled, torsions flanking pivot atoms are solved for analytically to ensure that the segments properly connect to one another and to the rest of the structure [ 19 ]. Generating a full conformational ensemble for an 8-residue macrocycle can require sampling up to 50,000 conformations [ 20 ]. However, this stochastic method is computationally intensive, often spending time on energetically unfavorable regions of the conformational landscape. More information on GenKIC can be found in Section 3.1 . Here, we present our model, which we call CyclicCAE (short for the Cyclic C onformational A uto E ncoder). This is a novel convolutional autoencoder architecture, capable of efficiently generating energetically favorable heterochiral macrocycle backbones. When trained on a set of conformational samples produced using GenKIC, CyclicCAE permits new conformations to be sampled from the distribution of accessible, low-energy closed conformations in a manner that is less computationally intensive than GenKIC. The autoencoder architecture, which permits inference at far lower computational expense than would be required for a diffusion model, was chosen to maximize the speedup. Because the training data may be generated once but a trained CyclicCAE instance may be used many times for many drug design problems, this translates into a considerable reduction in the computational cost of macrocycle drug design. CyclicCAE also offers versatility advantages over GenKIC, permitting in-painting against desired motifs, or context-sensitive gradient-descent refinement of backbone conformation on the manifold of accessible conformations, making it highly suitable for macrocycle drug design within the context of binding pockets on target proteins. The loss function used for training ensures that conformations with low root mean squared deviations (RMSDs) are mapped to points nearby in the latent space, facilitating local conformational searches to diversify from a promising starting point. We believe that CyclicCAE will be useful for reducing the computational cost of both producing initial candidate designs and of validating designs by conformational sampling, thus accelerating macrocycle drug design. 2 Results and Discussion 2.1 Dataset Construction Compared to linear peptides and proteins there are a limited number of experimentally determined heterochiral macrocyclic peptide structures or sequences. This makes it challenging to gather a large enough dataset for supervised or unsupervised training. Taking this into account, we generated 50,000 8-mer macrocycle conformations using Rosetta’s simple_cycpep_predict application [ 19 , 20 ], which acts as a wrapper for GenKIC to sample possible macrocycle conformations (see section 3.1 ). We used glycine’s Ramachandran potential to sample the mainchain torsion angles of each residue. Glycine is unique among the canonical α-amino acids because it is achiral, and the backbone dihedral angles accessible to glycine are roughly the union of the sets of backbone conformations accessible to L- and D-amino acids, as shown in Fig. 1A , left. Download figure Open in new tab Figure 1: ( A ) Comparison of torsion angles sampled by GenKIC (50,000 samples) in the dataset used to train the CyclicCAE model (left), and 50,000 random points sampled from the latent space of the trained CyclicCAE which were decoded (right). The distributions closely match one another, and reflect the Ramachandran preferences of glycine. ( B ) Full initila and fine-tuning training of CyclicCAE for 6,000 epochs. ( C ) Zoom in on the final fine-tuning training done from epochs 3000–6000. A custom scheduler was implemented with warm restarts and decreasing learning rates on plateaux. This is why the spikes are observed. In order to sample the conformational space of an octomeric peptide exhaustively we generated 50,000 conformational samples using simple_cycpep_predict [ 19 , 20 ]. The generated structures were then subjected to geometry optimization using RosettaQM, a bridge that is currently under development (manuscript in preparation) connecting Rosetta with the GAMESS quantum chemistry package [ 29 , 30 ]. Energies were then computed with both Rosetta’s ref2015 force field and with GAMESS using the DFTB semi-empirical quantum chemistry method [ 31 ]. To permit efficient training, we wrote a Rosetta application, extract_dof_vectors , which converts Rosetta structures to a binary database of records for input into a neural net. Each record included the ϕ, ψ, ω , mainchain torsion angles, mainchain bond angles and bond lengths, and the energy of each structure as scored using Rosetta’s ref2015 force field [ 32 ], the DFTB semi-empirical method [ 31 ], and the DFTB method with the fa_sol , fa_intra_sol_xover4 , and lk_ball_wtd implicit solvation terms from Rosetta’s ref2015 added. Notably, the training dataset does not include interatomic distances (or proxies like contact maps) for nonbonded atoms. 2.2 Model Training and Dataset Reconstruction We trained CyclicCAE for two rounds of training, each 3,000 epochs, where the first round was of initial training on k -clustered samples and the second round was fine-tuning with all conformers in the training conformational ensemble. A custom warm-restart scheduler was used during training, to help overcome getting stuck in local minimias. During training, a latent space loss term was used to enforce confermers spacing within the latent space to be similar to RMSD distances ( equation 13 ). Following the initial training and fine-tuning steps, explained in detail Methods Section 3.8, loss was evaluated for overtraining, and nice consistency between the training and validation loss were observed ( Fig. 1B – 1C ). We can see from the training graphs that an initial train and validation loss of 154,666 and 148,553, respectively, were observed. While a final loss of 2,789 and 2,708 were observed at epoch 6,000 for train and validation, respectively, confirming that our train and validation splits did not diverge from each other during training. We wanted to confirm that CyclicCAE effectively learned a realistic representation of the poly-glycine backbones in our input dataset. To verify this, we sampled 50,000 random points from the latent space and decoded them. The resulting ϕ and ψ angles were then extracted and compared to the input data. This allowed us to confirm that CyclicCAE accurately samples the full ϕ and ψ range of [ − 180, 180] while also retaining the Ramachandran preferences for glycine, as represented in the torsion angle distribution of the training dataset ( Fig. 1A , right). As one would expect, a considerable proportion of both the samples in the initial training dataset and those from the trained model clustered within the left- and right-handed α -helix regions of the Ramachandran plot. A 2D Earth Mover’s Distance metric of 0.0015 was calculated between the ( ϕ, ψ ) GenKIC dataset and CyclicCAE outputs. This indicates that the two distributions are quite similar, meaning CyclicCAE is sampling from a similar distribution as our base GenKIC dataset. 2.3 Correcting Terminal Bond Prediction Initial tests of CyclicCAE generated several macrocyclic peptides with stretched terminal bonds ( Fig. 2A – 2B , red distribution), an unrealistic outcome. Notably, however, the distribution had a peak around the standard peptide bond length of 1.4 Å [ 33 ], with a tail stretching to longer lengths. CyclicCAE only computes degrees of freedom (DoFs). In a closed chain, there is necessarily a breakpoint, the geometry of which is dependent on the other DoFs; as such, the terminal bond is not included in the DoF vector computed by CyclicCAE. To measure the terminal bond length, we therefore used the CyclicCAE model’s Ω(·) function (detailed in Section 3.4 ). Ω(·) computes the Cartesian coordinates for each atom within the backbone. Download figure Open in new tab Figure 2: Correcting terminal bond geometry in cyclic backbones produced by CyclicCAE. ( A ) Example of conversion of a stretched terminal bond to a corrected peptide bond of normal length. ( B ) Histogram of terminal bond lengths produced from 10,000 outputs with (green) and without (magenta) the terminal bond constraint. ( C ) Ramachandran contour plot demonstrating that the distribution of torsion angles sampled from CyclicCAE is similar without (top, magenta) and with (bottom, green) the terminal constraint, meaning there are no unnatural effects placed on the mainchain torsions ( ϕ,ψ ) by the terminal constraint. ( D - G ) Polar histogram plots showing the distribution of terminal peptide dihedral C α CNC α and bond angles ∠CNC α , ∠NCO, and ∠NCC α with (green) and without (magenta) the terminal term. Our terminal term not only generates outputs that are within an acceptable peptide length, but also guides our dihedrals and bond angles to ideal angles. To correct the streteched terminal bond, we introduced a loss term, ℒ Terminal Bond ( equation 3 ), in both the training and inference phases to enforce natural bond lengths (discussed in Section 3.4 ). Additionally, loss terms were added during inference for regressing terminal dihedrals (HNCO, C α CNC α ) and terminal bond angles (∠HNC, ∠CNC α , ∠NCO, ∠NCC α ) to their ideal angles. In order to test the effectiveness of our loss terms, we generated 10,000 samples with and without these terms. The addition of this term greatly tightened the distribution of bond lengths observed, shifting the mean from 2.54 ± 1.90 Å without the term (magenta in Fig. 2B ) to 1.40 ± 0.16 Å with the term (green). Next, we assessed whether the Ω( · ) adjustment maintained the torsion angles within their glycine-specific regions. Plotting the ϕ and ψ angles for both datasets ( Fig. 2C ) revealed similar distributions, indicating that while Ω(·) adjusts terminal bond lengths, it does not alter the torsion angles in ways that would deviate from glycine’s typical conformational preferences. In contrast, correcting terminal bond lengths using Rosetta’s gradient-descent minimizer would likely distort torsion angles, pulling them out of their preferred Ramachandran regions. Finally, we investigated whether the terminal bond dihedrals produced by CyclicCAE were within their acceptable range. The HNCO dihedral (not shown) is ideally 180 ° . With the terminal bond term, the distribution of dihedral values observed had a mean of 179.52 ± 24.48 ° , compared to 178.22 ± 55.56 ° without the terminal bond term. Dihedral C α CNC α ( Fig. 2D ) is also ideally 180 ° ; the terminal bond term yields a mean of 179.88 ± 15.78 ° , compared to 181.01 ± 41.65 ° without the term. Both dihedrals have properly centered distributions, with or without the terminal bond term. However, the standard deviation of those values is larger for without the terminal bond term. Terminal bond angles see larger improvements on addition of the terminal bond term: ∠HNC (not shown) has an ideal angle of 119 ° , and addition of the terminal bond term raises the observed bond angles’ mean from 113.30 ± 31.45 ° without the term to 116.91 ± 11.13 ° with the term, simultaneously tightening the distribution. The ideal ∠CNC α ( Fig. 2E ) is 120 ° , and we observe a mean of 103.26 ± 33.07 ° without the term compared to 114.54 ± 10.59 ° with the term. The ideal ∠NCO ( Fig. 2F ) is 123 ° , and we observe a mean of 114.02 ± 31.55 ° without the term compared to 118.38 ± 12.06 ° with the term. Lastly, the ideal ∠NCC α ( Fig. 2G is 116 ° , and our datasets have a mean of 99.35 ± 33.32 ° without the term and 111.91 ± 11.32 ° with the term. Each terminal bond angle’s distribution moves closer to the ideal angle, showing less variation, when the terminal bond term is included. 2.4 Gradient-Descent Energy Minimization for Geometry Relaxation Rosetta offers a minimization feature that, given a structure, an objective function f ( · ) (typically the Rosetta energy function), and the gradient of the objective function with respect to degrees of freedom of the system ∇f ( · ), carries out gradient-descent minimization of the objective function f ( · ) to relax the structure to a more favorable energy level [ 34 ]. Since a key objective of our approach is to take an input structure and minimize it energetically to another structurally similar conformation that is also part of the set of favorable conformations accessible to macrocycles, we incorporated energy as an output feature from CyclicCAE, and trained CyclicCAE to predict the energies of structures in the training set. Because neural nets like CyclicCAE are intrinsically differentiable, the inclusion of energy as an output permits gradient-descent minimization of energy in the latent space during inference, and also has the effect of shaping the latent space during training, which further facilitates energy minimization for geometry relaxation during inference, as detailed in Section 3.6.1 . To verify that minimized structures from the latent space continued to represent the full conformation space accessible to poly-glycine macrocycles, we sampled 10,000 points from the latent space and decoded them. We then used gradient marching to minimize the energy predicted by CyclicCAE with respect to latent space coordinates, starting from these same 10,000 points, over 1,000 steps. A 2D Earth Mover’s distance of 0.027 was calculated between the Ramachandran distributions (the values of backbone dihedral angles ϕ and ψ ) computed with and without gradient-descent energy minimization, indicating they are very similar. The energy-minimized samples (not shown here) shows a slightly stronger preference for the left- and right-handed α -helical regions as compared to the unminimized samples. This suggests that the most energetically favorable structures may reside within a more restricted subspace, V⊂P , of the complete torsion angle space. Each backbone DoF (bond lengths, bond angles, and dihedral angles) is being predicted by CyclicCAE, which allows us to place a harmonic constraint on any DoF we would like retained. This allows constrained gradient descent minimization in which we retain some desired DoFs, but sample freely the other DoFs. A pseudo-inpainting task is done using this approach, which is explored more thoroughly in Section 2.7 . We note that, while the α -helical regions are nearly symmetric, they lack perfect symmetry. This reflects the fact that CyclicCAE architecture did not have any sort of expectation of symmetry built into it, and had to learn that the conformational preferences of glycine are symmetric from the training data. While this indicates that additional samples or loss terms may be necessary to coax CyclicCAE to learn to sample in a perfectly symmetrical manner, we opted to keep CyclicCAE and its training process versatile enough to permit it to be trained on chiral chains in the future. 2.5 Markov Chain Monte Carlo Since one of the goals of our CyclicCAE model was to enable local conformational searching from an input structure, we implemented Markov Chain Monte Carlo (MCMC) as a method for exploring the latent space, as described in Section 3.6.2 . Although Markov Chain Monte Carlo is traditionally used to construct thermodynamic ensembles of physical systems [ 35 ], it has proven to be a useful means of searching large conformational spaces to find low-energy states independent of the distribution of states ultimately sampled [ 34 ]. Our implementation involves taking a random step in the latent space drawn from a Gaussian distribution of breath σ , evaluating a loss function at the new latent-space coordinate, and accepting or rejecting the move based on the Metropolis criterion. Because CyclicCAE’s training loss function ensures that similar structures remain close together in the latent space, we were able to test the effectiveness of our MCMC approach for local searches by decreasing the value of σ , or for more global searches by increasing the value of σ . By varying σ and comparing the RMSD of all generated structures (All vs All RMSD; Table 1 ), we observed that as σ increases, so does the structural diversity within the samples. Using these same outputs, we examined the proximity of each structure to the original input (All vs Native RMSD; Table 1 ). As expected, as the σ increases, the generated structures diverge further from the input structure. View this table: View inline View popup Download powerpoint Table 1: Comparison of samples generated by CyclicCAE over a 100 step MCMC trajectory as we vary the size of the step ( σ ). We compare both the variation among the samples that we draw over the MCMC trajectory (All vs All), and the variation of the samples from the input structure used as the starting point (All vs Native) 2.6 Compute Time Rosetta’s native implementation of GenKIC is the current state of the art for generating ensemble conformations for macrocyclic peptides [ 19 , 20 ]. We used GenKIC as a benchmark for comparing compute times in generating samples. Specifically, GenKIC was paired with MinMover as a localized minimizer within Rosetta, while CyclicCAE randomly sampled from the latent space, followed by 100 steps of gradient-descent energy minimization in the latent space. For both GenKIC and CyclicCAE, we generated 10, 100, 1,000, and 10,000 samples on a single CPU, with an additional GPU comparison for CyclicCAE. (Note that GenKIC does not support GPU computation, so no direct comparison was possible on this hardware.) As shown in Fig. 3A , CyclicCAE initially takes longer than GenKIC for generating 10 to 100 samples. However, at 1,000 samples, CyclicCAE begins to slightly outperform GenKIC, and at 10,000 samples, CyclicCAE shows a marked improvement: CPU-based CyclicCAE is 11 times faster, and GPU-based CyclicCAE is 33 times faster than GenKIC. Download figure Open in new tab Figure 3: ( A ) Comparison of the time (CPU- or GPU-seconds) needed to generate 10,000 8-residue macrocycle backbone conformations using GenKIC on CPU (orange), CyclicCAE on CPU (green), or CyclicCAE on GPU (blue). ( B ) Inpainting example starting from a native peptide macrocycle (opaque). A two-residue motif (cyan) was constrained to the input conformation, and conformations of the rest of the macrocycle (magenta) were sampled by inpainting. Four of the 5,000 sampled conformations from CyclicCAE are shown in transparent overlay (lime). ( C – E ) CyclicCAE histogram of outputs from inpainting with a 1 residue motif ( C ), 2-residue motif ( D ), and 3-residue motif ( E ) showing both motif (cyan) and scaffold (magenta) RMSD distributions compared against the input. This is understandable in terms of modern computing hardware: parallel computing capabilities, particularly on GPU, permit thousands of evaluations of a neural net at comparable computational cost to a single evaluation. These trends are also promising, since backbone conformational sampling when designing peptide drugs routinely involves thousands of samples per design, and conformational sampling simulations for validation can involve hundreds of thousands or even millions of samples per design [ 16 , 17 ]. Additionally, design pipelines rely on generating and validating large numbers of designs per target to maximize the likelihood of finding a short list of high-quality designs worth synthesizing and evaluating experimentally. When comparing these methods, our objective was not only to confirm CyclicCAE’s speed advantage, but also to assess its efficiency in generating stable structures. Compared to GenKIC, CyclicCAE consumes significantly less time and compute resources to generate poly-glycine backbones, as shown in Fig. 3A . 2.7 Conformational Motif Inpainting Conformational motif inpainting [ 36 , 37 ] involves sampling conformations of residues peripheral to a region of a macromolecule that is in a desired, fixed conformation ( Fig. 3B ). Traditional conformational motif inpainting is done on Cartesian atom coordinates, whereas our DoF based motif inpainting uses the torsion angles ( ϕ, ψ, ω ), bond angles, and/or bond lengths instead. Our approach is described in Section 3.6.3 . To test our DoF based conformational motif inpainting method, we took three contiguous motifs of an input macrocycle scaffold, ranging in size from one to three residues, and sampled conformations of the rest of the scaffold compatible with the fixed motif. We generated approximately 5,000 outputs for each of the three differently sized contiguous motifs. To assess the extent to which motifs were held fixed and to which we achieved diversity in the sampled scaffold conformations, for each ensemble of outputs, we measured the RMSD of the output motifs from the input, and of the output scaffold regions from the input. The first motif, which consisted of only a single residue ( Fig. 3C ), is similar to a traditional single-residue anchor starting point in peptide macrocycle drug design [ 24 , 25 ]. It took CyclicCAE 15s on a GPU to match and energy minimize for 100 steps our 5,000 poly-glycine backbone outputs. The mean RMSD of the motif compared to native was 0.37 ± 0.25 Å, with a minimum RMSD of 0.02 Å and a maximum RMSD of 0.65 Å, indicating that this motif was appropriately immobilized. In contrast, the scaffold RMSD compared to native was 2.18 ± 0.39 Å, with a minimum RMSD of 0.47 Å and a maximum RMSD of 3.13 Å, indicating considerable diversity in the conformational sampling, as desired. For the second motif, which consisted of two contiguous residues ( Fig. 3D ) it took CyclicCAE 26s on a GPU to again match motifs and energy minimize for 100 steps our 5,000 poly-glycine backbone outputs. From the outputs generated by CyclicCAE the mean motif RMSD when compared against native was 0.13 ± 0.07 Å with a minimum of 0.03 Å and a high of 0.67 Å, again indicating complete immobilization. In contrast, the scaffold region when compared to native had a mean RMSD of 1.70 ± 0.40 Å, with a minimum of 0.43 Å and a maximum of 2.60 Å, again indicating considerable diversity in the conformational sampling. The final test, which consisted of a three residue contiguous motif ( Fig. 3E ), which took CyclicCAE 1 min & 12s to match motifs and energy minimize for 100 steps our 5,000 poly-glycine backbone outputs. Within the outputs the motif mean RMSD when compared to native was 0.34 ± 0.08 Å, where the minimum RMSD was 0.17 Å and the maximum RMSD was 0.73 Å, again indicating complete immobilization. The scaffolds when compared to native had a mean of 1.73 ± 0.43 Å, with a minimum RMSD of 0.29 Å and a maximum RMSD of 2.56 Å, once more suggesting extensive conformational sampling. These results reaffirm our previous findings, highlighting the versatility of our approach for motif-specific scaffold design. They additionally demonstrate the speed advantage one gets when using CyclicCAE to generate motif-scaffolded heterochiral poly-glycine macrocycle backbones. 2.8 De novo Conformational and Sequence Design As a final test of our method’s validity against GenKIC, we sampled 10,000 new backbone conformations with both GenKIC and CyclicCAE. These 10,000 backbones were then subjected to a subsequent Rosetta sequence design process with one replicate per input backbone. These peptides were designed only to fold in isolation in aqueous solvent (not to bind to any target). Our design protocol (see Section 3.9 ), scripted in the RosettaScripts scripting language [ 38 ], started with an input backbone from either GenKIC or CyclicCAE (step 1) which was subjected to gradient-descent energy minimization with Rosetta’s MinMover using the ref2015 energy function (step 2). D- or L-proline residues were next placed at positions that were not donating backbone hydrogen bonds and which were in compatible regions of Ramachandran space using Rosetta’s flexible-backbone FastDesign protocol [ 19 ] (step 3). The rest of the sequence was then designed with FastDesign (step 4), and the structure was relaxed with Rosetta’s FastRelax protocol [ 39 ] (step 5), yielding the output structure on which final analysis was performed (step 6). In this final step, we performed several measurements to capture key characteristics of each dataset. These included identifying the number of internal backbone-backbone and sidechain-backbone hydrogen bonds within each structure ( Fig. 4A ). Download figure Open in new tab Figure 4: Comparison of full peptide design using CyclicCAE or GenKIC starting scaffolds. ( A ) Internal hydrogen bond counts for designs from CyclicCAE (blue) or GenKIC (orange) starting scaffolds. Frequencies for total internal hydrogen bonds (left), sidechain-backbone hydrogen bonds (center), and backbone-backbone hydrogen bonds (right) are shown. CyclicCAE scaffolds had considerably more internal backbone-backbone hydrogen bonds, yielding designs with more total hydrogen bonds and fewer sidechain-backbone hydrogen bonds. ( B ) Rosetta ref2015 scores for designs from CyclicCAE (blue) or GenKIC (orange) starting scaffolds. CyclicCAE-based designs were considerably lower-energy on average. ( C ) Cumulative probability distributions showing fraction of designs with fold propensity ( P Near ) values less than or equal to a specified value. Better methods yield more designs with higher P Near values, reducing the area under the curve. Four variant design scripts are compared using CyclicCAE (left) or GenKIC (middle) starting scaffolds. At right, the design script yielding best results is compared for CyclicCAE (blue) and GenKIC (orange) starting scaffolds. When comparing total internal hydrogen bonds (left), GenKIC designs have a mean of 3.72 ± 1.30 hydrogen bonds per design, while our CyclicCAE’s designs a have mean of 4.84 ± 1.21 total internal hydrogen bonds. Unsurprisingly, CyclicCAE is better able to focus on heavily hydrogen-bonded regions of conformational space, which are more likely to yield energetically-favorable conformations. Interestingly, GenKIC designs had on average more sidechain-backbone hydrogen bonds than did CyclicCAE designs (2.29 ± 1.06 hydrogen bonds for GenKIC vs . 1.24 ± 1.26 for CyclicCAE). This is likely due to the fact that a large proportion of macrocycles generated by GenKIC have unsatisfied hydrogen bond donors and acceptors prior to sequence design, which increases the likelihood for subsequent sidechain placement that creates sidechain-backbone hydrogen bonds, since the FastDesign process tries to minimize the energy of the design. GenKIC backbones exhibit a mean of 1.43 ± 1.18 backbone-backbone hydrogen bonds, while CyclicCAE designs have a mean of 3.49 ± 1.26 backbone-backbone hydrogen bonds ( Fig. 4A ), reinforcing CyclicCAE’s capacity to increase macrocycle stability through internal hydrogen bonds [ 24 ]. We also evaluated each design by scoring it using the ref2015 Rosetta energy function [ 32 ]. As shown in ( Fig. 4B ), designs produced by both methods have similar score distributions, but the CyclicCAE’s distribution is shifted to significantly lower energies, indicating an increased propensity to produce more stable structures. CyclicCAE designs have a mean score of -8.091 ± 6.38 kcal/mol, whereas GenKIC designs have a mean of -6.292 ± 5.41 kcal/mol. Following this, we filtered out peptides with scores > 0 kcal/mol, resulting in 8,713 GenKIC designs and 8,663 CyclicCAE designs. From these peptides, we randomly selected 1,000 from each design pool (CyclicCAE and GenKIC) for folding using simple_cycpep_predict [ 19 ], which applies GenKIC to the sequences, allowing us to calculate the fold propensity metric P Near ( equation 1 ) [ 24 , 19 , 20 , 40 ]. P Near , which ranges from 0 to 1, is an approximation of the peptide’s fractional occupancy in the designed conformation. A P Near value of 0 implies no time in the designed conformation, while a P Near of 1 indicates full occupancy. In equation 1 , N is the number of sampled conformations, E i and E j are the energy of the i th and j th samples, respectively, and k B T is the Boltzmann temperature. For each sample, r i is the RMSD to a target conformation, and λ is a factor that determines how close a sample must fall to the target conformation in order to be considered “native-like”. For small peptides, a value of 1.5 Å is commonly used for λ . Fig. 4C plots the cumulative distribution (CD) for P Near scores to evaluate each of our different design method. The CyclicCAE backbones consistently showed a greater propensity of achieving high P Near scores (left) compared to GenKIC backbones (middle). In the final design method (Non-ideal/Non-Cartesian; see Section 3.9 ), the P Near CD ( Fig. 4C right) clearly demonstrates a larger proportion of higher P Near peptides generated by our CyclicCAE than by GenKIC. Given that CyclicCAE is implicitly rather than explicitly conditioned by Rosetta’s energy function, we examined Rosetta’s impact on backbone perturbation during major design steps. There are three primary design stages: proline addition, sidechain design, and final relaxation. To introduce minimal Rosetta optimization to our backbones, we applied the MinMover (Rosetta-based gradient-descent energy minimization, [ 34 ]) at the beginning of the design script ( Section 3.9 ) to both GenKIC and CyclicCAE backbones. The RMSD was calculated between the initial input and the MinMover-optimized backbone. GenKIC backbones stayed close to their initial conformation throughout the design process, while CyclicCAE designs remained close but shifted to slightly higher RMSD values ( Fig. 5B , top). Comparing each step to MinMover-optimized structures reveals smaller deviations and more overlap, as in smaller conformational movements, between the two backbone methods. Download figure Open in new tab Figure 5: ( A ) Rosetta design workflow carried out on backbones generated by GenKIC or CyclicCAE. ( B ) Backbone changes over the course of the design protocol. In each case, the backbone heavy atom RMSD is computed. (Top) Backbone RMSD from the original input from GenKIC (orange) or CyclicCAE (blue) after various steps in the design protocol. (Bottom) Backbone RMSD from the relaxed input structure produced by the MinMover after various steps in the design protocol. ( C ) Structure comparison for a representative CyclicCAE poly-glycine backbone yielding a high fold propensity ( P Near ) design prior to (magenta) and and following (cyan) the Rosetta design protocol. Relaxation by the MinMover, FastDesign, and FastRelax cause relatively small changes in backbone conformation, and preserve most of the hydrogen bonding pattern produced by CyclicCAE. ( D ) Structure comparison for a representative GenKIC poly-glycine backbone yielding a high fold propensity ( P Near ) design prior to (magenta) and following (orange) the Rosetta design protocol. Relaxation by the MinMover, FastDesign, and FastRelax result in significant backbone conformational change, turning a relatively open structure produced by GenKIC into a heavily hydrogen-bonded structure. ( E ) Average compute time to reach a desired fold propensity ( P Near ) threshold for (left) CyclicCAE, (middle-left) GenKIC, or (middle-right) CyclicCAE (blue) and GenKIC (orange) compared. At right, the ratio of GenKIC to CyclicCAE compute time to achieve a desired P Near value is plotted. This analysis suggests that pre-minimizing CyclicCAE backbones with MinMover could be optimal for better alignment with Rosetta’s force field prior to further design. Despite starting slightly out of the energy minima for the Rosetta ref2015 energy function, CyclicCAE designs had a much higher average fold propensity ( P Near ). Given the large number of designs from GenKIC with P Near values near zero, it appears that GenKIC-generated poly-glycine backbones may not be ideal starting points for design in comparison ( Fig. 5C-5D ). We next assessed the mean amount of time it takes to generate a successful high P Near structure with CyclicCAE compared to GenKIC. CyclicCAE takes nearly two orders of magnitude less time to generate high P Near structures as it does for GenKIC. If one’s target is a design with a P Near ≥ 0.8, CyclicCAE achieves this in 70-fold less time than GenKIC; if one’s target is P Near ≥ 0.9, CyclicCAE beats GenKIC by a factor of 80. 2.9 A Proof of Principle for Using CyclicCAE in Drug Design and Validation Pipelines In order for CyclicCAE to be useful, it must be able to efficiently sample macrocycle conformations relevant to real-world use cases. Two common use cases arise when designing macrocycle drugs intended to bind to a target of therapeutic interest. First, during the design phase, one wishes to rapidly sample plausible backbone conformations for subsequent sequence design. Second, during the validation phase, one wishes to sample all plausible backbone conformations of a given designed macrocycle sequence in order to test whether a desired, binding-competent conformation represents a unique low-energy state [ 16 , 17 , 18 , 40 ]. To test whether CyclicCAE can serve as a useful backbone conformational sampler for these two use cases, we examined the model’s ability to efficiently sample close to the conformation of an established peptide macrocycle, NDM1i-1G. This molecule was previously reported to bind and to inhibit the New Delhi metallo- β -lactamase 1 (NDM-1) with an IC 50 value of 1.2 ± 0.1 µ M [ 24 ]. NDM-1 is an enzyme used by certain pathogenic bacteria to degrade β -lactam antibiotics, including last-resort drugs like carbapenems, making it an important drug target [ 41 , 42 ]. As a proxy for CyclicCAE’s use for design or validation in a de novo drug design project, we tested CyclicCAE’s ability to recapitulate NDM1i-1G’s backbone conformation through inpainting-based backbone generation, using one of the NDM-1 crystal structures available in the PDB (6xbf). Following the original design approach, we began with L-captopril, a small molecule inhibitor of NDM-1, and converted this to a dipeptide to use as a fixed motif for inpainting [ 43 , 44 , 45 , 24 ]. We generated 5,000 poly-glycine backbones from this motif using CyclicCAE, minimizing each structure with CyclicCAE’s gradient marching energy minimization; on a single GPU, this took 26 seconds. Of these, 40 (0.8% of the total) had an RMSD ≤ 0.8 Å from the NDM1i-1G peptide’s conformation in the 6xbf crystal structure, with the closest sample showing an RMSD of only 0.423 Å. To confirm that CyclicCAE is sampling adequately closely to the physical conformation for use in design or validation pipelines, we threaded the NDM1i-1G sequence onto the backbone and relaxed the macrocycle in isolation ( i . e without the target) using Rosetta’s FastRelax protocol. We found that after relaxation the structure remained close to the experimentally-determined structure, with backbone heavy RMSD values that ranged between 0.3 and 0.95 Å ( Fig. 6A ), indicating that CyclicCAE had successfully sampled deeply enough in the lowest-energy well in the conformational landscape for downstream sidechain packing and energy minimization algorithms to remain in this well, and in many cases to move closer to the native conformation during relaxation. A representative sampled, relaxed conformation generated using this method (green) is shown overlain on the NDM1i-1G x-ray crystal structure (pink) in the context of the NDM-1 target (grey) in Fig. 6B . Download figure Open in new tab Figure 6: ( A ) RMSD comparison of our CyclicCAE outputs to the conformation of the NDM1i-1G peptide in PDB structure 6xbf. (left) Distributiuon of RMSD values for 5,000 samples. 40 samples (0.8% of the total, cyan) fell within 0.8 Å RMSD of the experimentally-observed conformation. (middle and right) All heavy-atom RMSDs (middle) and backbone heavyatom RMSDs (right) of samples with RMSD values ≤ 0.8 Å after threading the NDM1i-1G sequence, packing side-chains, and relaxing the conformation using Rosetta’s FastRelax. ( B ) One representative poly-glycine backbone from CyclicCAE with the NDM1i-1G sequence threaded and the conformation relaxed (green) overlain on the NDM1i-1G crystal structure (pink and grey). A backbone heavyatom RMSD of 0.318 Å was observed. ( C ) The L-captopril based dipeptide motif (cyan) with scaffold region sampled by CyclicCAE (green) is shown prior to (left) and after (right) gradient-descent minimization including a Lennard-Jones term in the context of the NDM-1 active site pocket. Relaxation relieves clashes without tearing apart the favorable hydrogen bonding pattern. ( D ) Effect of inclusion of the Lennard-Jones term during conformational sampling and relaxation. The fraction of generated backbones lacking clashes with a tight GPCR target pocket are shown without (left) and with (right) the Lennard-Jones term. Clash distance thresholds of 0.5 Å (light blue) and 1.0 Å (dark blue) were used. The Lennard-Jones term greatly increases the fraction of samples that are clash-free, providing a rapid means of sampling macrocycle conformations that fit into a target binding pocket. 2.10 Sampling Macrocycle Backbone Conformations in the Context of a Target Binding Pocket In the examples above, CyclicCAE was used to sample macrocycle conformations in isolation, with no consideration of the macrocycle’s surroundings. However, macrocycle drug design requires that one sample plausible backbone conformations in the context of a binding pocket on a target protein, in order to find backbones that could likely be stabilized by suitable subsequent choice of sequence. In order to enable macrocycle design for a particular interface/pocket, we added a Lennard-Jones style repulsion term, explained in Section 3.7 , to guide CyclicCAE when sampling macrocycle conformations. By providing CyclicCAE with a target protein structure and by placing interface interacting residues to be used as a motif, we can sample macrocycle conformations that are both stable in isolation and shape-complementary to a provided target. After sampling a conformation by drawing a point randomly from the latent space, the inclusion of the Lennard-Jones term during gradient-descent minimization in the latent space guides CyclicCAE to produce backbone conformations that do not clash with the target. Because minimization occurs in the latent space, the structures produced retain the quality of being stable, low-energy conformations, often rich in favorable features like intramolecular hydrogen bonds – something that is not true when carrying out gradient-descent energy minimization in torsion space, where lever-arm effects can tear hydrogen bonds apart or cause major distortion to structures in order to relieve clashes. This process is described in detail in Sections 3.6.3 and 3.7 . CyclicCAE pocket-context minimization is illustrated in Fig. 6C . As a test, we generated two sets of 500 macrocycles. One set was generated in isolation ( i . e . with no pocket context) and another within the context of a target protein binding pocket. For this task, we used two targets: NDM-1, described in Section 2.9 , and a G-protein coupled receptor (GPCR). The latter was used both since GPCRs are a class of protein of high interest as drug targets, and since they have tight pockets that make it particularly challenging for existing methods to generate stable macrocycles that do not clash with a GPCR’s sidechains or backbone atoms. When backbones were generated and relaxed by CyclicCAE without knowledge of the pocket context, only 1.8% had by chance no clashes with the target, based on a requirement that all backbone atoms be at least 0.5 Å from any target atom. When the stringency of this requirement was increased, so that the clash distance threshold was 1.0 Å, then 0% of the generated backbones lacked clashes. In contrast, including the pocket context (and the Lennard-Jones term) during relaxation by CyclicCAE resulted in 96.6% of generated backbones lacking clashes with the target given the 0.5 Å clash threshold, and 66.4% lacking clashes given the 1.0 Å threshold ( Fig. 6D ). Note that this was based on a static representation of a GPCR, and did not account for the repacking of sidechains that could possibly relieve some clashing. Nevertheless, this is a clear demonstration of the efficacy of CyclicCAE for rapidly finding macrocyclic backbones compatible with any given target binding pocket. 3 Methods 3.1 Generalized Kinematic Loop Closure (GenKIC) Generalized Kinematic Closure, or GenKIC, is the current state of the art method for generating heterochiral macrocycle poly-glycine backbones in silico . This method is a part of the Rosetta macromolecular modeling suite, and is based on a kinematic approach used in the robotics field that was adopted for loop conformation sampling [ 27 , 28 ]. One can think of a macrocycle as a set of N main chain atoms forming a closed loop, with N dihedral angles, N bond angles, and N bond lengths defining the degrees of freedom of the system. If the atoms of one residue (in the case of α -amino acids, these would be the three N, C α , and C atoms) are removed to define a fixed frame of reference, the remaining N − 3 (or, in the general case of D main chain atoms per residue, the remaining N− D atoms) form the loop to sample. In this loop, any given conformation is defined by specifying all of the dihedral angles, bond angles, and bond lengths of the loop. However, given ideal bond angles and bond lengths, the known rigid-body transform from the start of the loop to the end (to ensure that it connects back to the excluded residue) reduces the effective number of degrees of freedom by 6. This means that one can sample, perturb, or randomize all but 6 dihedral angles in this chain and solve analytically for the remaining 6 to impose the condition that the chain remain closed. Because solving the kinematic equations is extremely fast, this was a much less computationally expensive means of rapidly generating closed macrocycle conformations than more traditional approaches like molecular dynamics simulations. However, while the sampled degrees of freedom could be biased by the Ramachandran preferences of the residues in the chain (in our case, glycine), the six degrees of freedom that were solved for analytically could end up taking any values. Post-hoc filtering was needed to discard solutions in unreasonable regions of Ramachandran space, or with other undesirable features such as a paucity of intramolecular backbone hydrogen bonds. This necessitated far more sampling in order to find the energetically favourable conformations. Where the original KIC approach in Rosetta was hard-coded to operate on α -amino acid backbones, GenKIC was written, as the name implies, to be general, permitting closure of any chain of atoms. It could operate on any mixture of α -, β -, or γ -amino acids, or on chains connected by side-chain linkages such as disulfide bonds or chemical crosslinkers [ 19 , 21 ]. 3.2 CyclicCAE Neural Network Architecture The poly-glycine macrocyclic peptide conformational space spans all possible L- and D-chiral structural designs (including all possible hetero- or homochiral sequences). By exhaustively sampling this comprehensive conformational space, we gain access to all potential heterochiral backbone structures. To convert GenKIC’s stochastic process into a continuous one, we introduce a mapping, η (·) : ℝ D → ℝ P , from the input dimension to the latent dimension, to map the conformational degrees of freedom (DoF) into a much reduced latent space ℝ P . Here, η (·) serves as an encoder neural network, while η − 1 (·) : ℝ P → ℝ D maps the latent space back to the input dimension. Our objective is to take an input DoF vector, x∈ ℝ D , and accurately reconstruct it. This mapping from input to latent space and back can be framed as an autoencoder [ 46 ], represented as: Since angles (such as torsion and bond angles) are components of the DoF, they present a unique challenge compared to linear terms. Due to the circular nature of angles, where -180 ° and 180 ° are adjacent, we cannot apply simple regression as with linear terms. To address this, angles are converted into radians, after which their sin(·) and cos(·) values are calculated. This approach allows regression or mean squared error (MSE) to be applied to the sine and cosine values, from which the original radian can be recomputed using atan2(·). Convolutional layers capture interactions between atoms within and between residues, whether those atoms are close or distant in space [ 47 , 48 , 49 ]. Dense layers are used to compress relevant information into a reduced representation. Residual connections allow important information from earlier steps to flow into the reduced representation, improving gradient flow and reducing computational cost [ 50 ]. One unique feature of our architecture was that the decoder arm, η − 1 (·), actually maps R P − > R D + Q , where Q is one or more energies predicted by the decoder (which were not present in the original inputs). During training, the loss function penalizes both deviation of conformational degrees of freedom from the inputs, as well as deviation of predicted energy values from known values (see Section 3.3 ). During inference, the network’s ability to predict energy values can be used to compute gradients of energy with respect to latent space coordinates, and to thereby relax structures by gradient-marching (see Section 3.6 ). 3.3 Loss Terms The loss functions outlined in the method sections below are integrated into two primary terms, ℒ Reconstruction and ℒ Latent Space Enforce , which are used during training ( equation 3 ). Inference employs a different set of loss terms, denoted as ℒ Inference , which includes a energy-specific term, I energy , which is a value outputted by CyclicCAE. A repulsion energy term, E repulsion , as described in Section 3.7 , along with a harmonic constraint for motifs of interest, ℒ Harmonic ( Section 3.6.3 ). The inference loss also incorporates certain training ℒ terms used in the training loop. Above, ℒ Angle,Dihedral Constraint includes some additional angle and dihedral constraints on the terminal bond. This is done by regressing to an ideal angle for each to ensure proper terminal peptide orientation and angles. 3.4 Constraining Terminal Bond Length The terminal bond length of a macrocycle is not supplied by a macrocycle’s DoFs, but instead is a result of threading the DoFs onto a macrocycle and computing the distance between the resulting N and C-termini atom placement. Therefore, we can compute the xyz coordinates of each atom within the poly-glycine macrocycle by applying the function Ω(·) to a macrocycle’s DoFs. With all atomic positions known, we can calculate the distance between N 0 and C L to determine our terminal bond length. A loss term, ℒ terminal bond ( equation 6 ), can then be introduced with a constant length c , 1.4Å, to enforce structures to approach a standard peptide bond length. To calculate the xyz coordinates per residue, we apply Ω(·), where residues are represented as a set , requiring CyclicCAE outputs: dihedrals ( ϕ, ψ, ω ), bond angle, and bond length information. We define functions R torsional (·) and R bond angle (·), which handle torsion angle rotations and bond angle rotations, respectively: We generate an atom’s xyz vector by iterating through each residue L i and each atom atom coordinates calculated is a hydrogen of L 1 , we use the initial nitrogen, , as the base, and an identity matrix for the rotation matrix R : Here, BL and BA denote bond length and bond angle, respectively, with subscripts identifying the atoms (e.g., BL NH is the bond length between nitrogen and hydrogen). Next, we update the rotation matrix R : With this updated R , we calculate the next atom, C α : With and the updated R matrix determined, we proceed to calculate : Finally, we calculate the coordinates for and of L i and L i +1 , respectively: This completes the atomic coordinates for L i , allowing us to proceed to L i +1 with the calculated starting atom, from L i +1 . 3.5 Positional Information Since macrocycles are circular with no distinct end, it would seem intuitive to use cyclic offset-based positional encoding. However, we want CyclicCAE’s latent space representation to change predictably under cyclic permutations, rather than treating all permutations as identical and mapping them all to the same point in the latent space. This distinction is crucial; for instance, when designing or minimizing scaffolds within a binding pocket, maintaining specific residue positions to interact with the pocket is essential. Taking that into consideration, we decided to apply linear positional encodings. 3.6 Inference Given that the latent space, ℝ P , is shaped both by structural and energetic information, this allows us to perform multiple inference based tasks such as energy minimization, Markov Chain Monte Carlo (MCMC) structural searches based on an input, and motif inpainting. 3.6.1 Energy Minimization To energetically minimize a macrocyclic peptide, one can encode it into the latent space using η (·) and perform gradient descent with respect to energy in this space. This process involves calculating the gradient with respect to latent space coordinates at a latent space point . One then marches in the negative gradient direction (the direction of decreasing energy) to successive points , decoding the structure using to recompute the energy. When energy ceases to decrease, one repeats the process, until a plateu is reached or until a user specified number of iterations, s has been exhausted. At the final point , the decoded latent space coordinate yields the DoF vector that defines the relaxed structure, as well as the corresponding predicted energy. 3.6.2 Markov Chain Monte Carlo Encoding a macrocyclic structure into CyclicCAE maps it to a specific point in the latent space. To generate structurally similar outputs, we can employ MCMC to explore the local region around this point in the latent space. Because during training our latent space is regulated by a structural enforcing term, ℒ Latent Space Enforce ( equation 13 ), similar structures in terms of RMSD should remain close to one another in latent space. Conversely, if we desire more structurally diverse outputs, we can use MCMC to sample distant regions on the manifold, far from the original point. 3.6.3 Motif Inpainting If a user wishes to preserve a specific anchor position or interaction while designing scaffolds, this can be achieved with CyclicCAE by minimizing ℒ Harmonic : To preserve a motif (or subset of residues) while designing other positions, we freeze the desired DoFs ( ϕ, ψ, ω , bond angles, and/or bond lenghts) and randomly sample the latent space for structures with similar DoFs to the selected residues. Starting with a predetermined number of scaffold structures, we can use MCMC to explore local structures and identify optimal starting structures, harmonic constraint ( equation 14 ) is used as a cutoff to determine if a structure contains the desired motif. Once an initial set of scaffolds is identified, we apply energy minimization for s steps, using ℒ Harmonic to maintain our DoFs of interest. This process both minimizes the structure and directs minimization to help preserve our desired motif. 3.7 Pocket Context Design CyclicCAE is not directly trained for minimizing structures within binding pockets. However, structure minimization can be guided by adding pairwise distance constraints between the peptide and the atoms in the pocket. We define a repulsion function, E repulsion (·), which applies a Lennard-Jones-like repulsion to the peptide in relation to the surrounding pocket atoms. Using a Lennard-Jones style energy term [ 32 ], as shown in equation 15 , we steer the minimization process: The parameters m and b are optimized for a strong repulsion when atoms are in close proximity, while distances within 0.6 σ < d ≤ σ follow a traditional Lennard-Jones repulsion potential. Distances beyond σ are not considered. To apply this, it is essential to align CyclicCAE-generated peptides onto the reference pdb structure since the peptide’s generated coordinates may differ spatially. We achieve this alignment using the Kabsch algorithm [ 51 ], aligning CycliCAE’s Cartesian output points with the pdb peptide points . First, we calculate the geometric centroids of both point sets: Then, we center these points around their respective centroids: The translation vector t ∈ ℝ 3 is obtained by computing the difference between the centroids: Next, we form a covariance matrix H ∈ ℝ N ×N and apply Singular Value Decomposition (SVD) to obtain U ∈ ℝ 3 × 3 , single value S ∈ ℝ 3 , and conjugate transpose rotation matrix V * ℝ ℝ 3 × 3 which minimizes the RMSD between P and Q . Using SVD, we decompose H : The rotation matrix R is calculated as: If det( R ) < 0, we adjust by flipping the sign of the last column in V before recalculating R . Finally, we align the peptide points P by applying the rotation R and translation t to produce the aligned points P a ∈ ℝ N × 3 : With this alignment, we compute the pairwise distances between CyclicCAE’s peptide atoms and the pocket atoms, allowing us to evaluate and optimize E repulsion as we proceed with steps in the latent space. 3.8 Model Training The CyclicCAE model was initially trained on a clustered subset of the input data, generated using Rosetta’s energy_based_clustering application [ 20 ]. The original dataset of approximately 50,000 inputs was clustered, and the 10 examples closest to each cluster center were selected, reducing the dataset to 1,280 samples. This subset was trained over 3,000 epochs with a learning rate of 1 × 10 − 3 , using 3 encoder and decoder layers, a noise factor of 5, 5 noise iterations, a dropout rate of 0.1, and a batch size of 128. After initial training, fine-tuning was conducted on the full dataset of approximately 50,000 samples for an additional 3,000 epochs with a learning rate of 1 e − 3 , a noise factor of 2, 5 noise iterations, dropout of 0.2, and a batch size of 2,064. During both initial training and the fine-tuning a custom scheduler was used to help overcome local minima. A custom function controlling the learning rate, which reduced the rate when validation loss had plateaued, was implemented. Warm restart logic increased the learning rate gradually back up to its initial starting value once a low threshold learning rate had been reached. This is the reason for the observed spiking behavior in Fig. 1C . 3.9 Rosetta Design The CyclicCAE model trained on poly-glycine backbones exclusively generates poly-glycine backbones, which then require side chain placement and energy minimization using Rosetta’s FastDesign and FastRelax, respectively. For a fair comparison between GenKIC and our CyclicCAE, we used the same design script to generate peptides intended to be stable in isolation (see Section 2.8 ). This design script comprises four six steps: an input from either CyclicCAE or GenKIC, backbone relaxation using Rosetta’s MinMover (which carries out gradient-descent minmization of the Rosetta ref2015 energy function), proline placement and relaxation, full sequence design with Rosetta’s FastDesign [ 19 ], final relaxation with Rosetta’s FastRelax [ 39 ], and final analysis of the output structure. These steps are illustrated in Fig. 5A . After generating backbones with either CyclicCAE or GenKIC, the MinMover was applied as an initial non-Cartesian ( i . e . torsion-space) minimization step. Since CyclicCAE structures are relaxed using an estimate of the Rosetta ref2015 energy provided by the neural net, and not based on actual Rosetta energies, this initial step acclimates the CyclicCAE-generated backbone to the Rosetta force field with slight dihedral adjustments. To ensure consistency, we also applied this initial minimization to GenKIC backbones, though these should require minimal adjustment as they were generated within Rosetta’s force field context (which can be seen in Fig. 5B . Following step two, our third step was to introduce proline residues at positions that were not donating backbone hydrogen bonds, and which were in regions of Ramachandran space compatible with either D- or L-proline. After mutating these positions to proline, we performed a round of FastRelax to relieve any strain that the introduction of proline may have caused. In our fourth step, FastDesign was applied across the macrocycle to select the residue identity at all non-proline positions, sampling side chains and rotamers with ideal dihedral angles and bond lengths. This step included minimization for all residues, repeated over 5 rounds. In step five, a single round of FastRelax was conducted, performing Cartesian minimization (permitting bond angles and dihedral angles to deviate slightly from ideal values) for final structural refinement. The sixth and final step was to measure properties of the design (such as hydrogen bond counts) and to write the final structure. 4 Conclusions In this work, we introduce the first general deep learning approach for generating de novo heterochiral macrocycle backbones, which we call CyclicCAE. We demonstrate the CyclicCAE model architecture’s functionality by training it to efficiently sample low-energy conformations of 8-mer poly-glycine macrocycles. Our work demonstrates that, having exhaustively sampled the conformation space of a class of macrocycle once using a slow but general sampling method (GenKIC), our CyclicCAE model permits much more rapid subsequent sampling of the most energetically favorable conformations, while also offering the versatility to permit efficient sampling in the context of target binding pockets, or to allow preservation of a desired binding motif while in-painting the rest of the macrocycle. Significantly, nothing about the model is specific for peptides of this size, for poly-glycine chains, or even for α -amino acids. Given suitable training data (from GenKIC-based sampling, MD simulations, experimental databases, or other sources), this general architecture can be used for any size of macrocycle built from any combination of building-blocks. We anticipate that CyclicCAE will be a powerful tool for reducing the computational cost of both designing new macrocyclic therapeutics, and for carrying out rapid conformational sampling simulations to validate designs, thus accelerating macrocycle drug discovery pipelines. 6 Conflict of Interest Statement VKM is a co-founder and shareholder of Menten AI, a peptide macrocycle drug design company. The other authors declare no conflicts of interest. 7 Code and Model Availability The authors plan to make the CyclicCAE model architecture, training scripts, and weights for CyclicCAE trained on particular training datasets available on a free and open-source basis, and are in the process of preparing and packaging CyclicCAE for publication. In the meantime, researchers interested in the approach should contact Parisa Hosseinzadeh ( parisah{at}uoregon.org ) and Vikram K. Mulligan ( vmulligan{at}flatironinstitute.org ). 5 Acknowledgments ACP thanks the Center for Computational Biology at the Flatiron Institute for hospitality while (a portion of) this research was carried out. The computations reported in this paper were (in part) performed using resources made available by the Flatiron Institute. The Flatiron Institute is a division of the Simons Foundation. PDR and VKM were funded by the Simons Foundation. ACP thanks the post-doctoral scientists Bargeen Turzo and Qiyao Zhu at the Flatiron Institue, and Kevin Harnden at the University of Oregon for their helpful conversations and providing feedback. ACP would also like to thank Erik Thiede who is a professor at the University of Cornell the Department of Chemistry and Chemical Biology for the helpful talks and ideas while working on this project. PH and ACP are funded through the NIH grant DP2GM146249f. 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