Sequence Space Research Articles

Retracting a ball in $\ell_1$ onto its simple spherical cap

In this article, a notion and classification of spherical caps in the sequence space $\ell_1$ are introduced, and the least Lipschitz constant of Lipschitz retractions from the unit ball onto a spherical cap is defined. In addition, an approximation of this value for the specific spherical cap, the simple spherical cap, is calculated. This approximation reveals a rough relation between these values, denoted by $\kappa(\alpha)$, and the answer of the optimal retraction problem for the space $\ell_1$, denoted by $k_0(\ell_1)$. To be precise, there exists $-1< \mu< 0$ such that $k_0(\ell_1)\leq\kappa(\alpha)\leq2+k_0(\ell_1)$ whenever $-1< \alpha< \mu$; here $\alpha$ is the level of spherical cap.

RrA, an enzyme from Rhodospirillum rubrum, is a prototype of a new family of short-chain L-asparaginases.

L-Asparaginases (ASNases) catalyze the hydrolysis of L-Asn to L-Asp and ammonia. Members of the ASNase family are used as drugs in the treatment of leukemia, as well as in the food industry. The protomers of bacterial ASNases typically contain 300-400 amino acids (typical class 1 ASNases). In contrast, the chain of ASNase from Rhodospirillum rubrum, reported here and referred to as RrA, consists of only 172 amino acid residues. RrA is homologous to the N-terminal domain of typical bacterial class 1 ASNases and exhibits millimolar affinity for L-Asn. In this study, we demonstrate that RrA belongs to a unique family of cytoplasmic, short-chain ASNases (scASNases). These proteins occupy a distinct region in the sequence space, separate from the regions typically assigned to class 1 ASNases. The scASNases are present in approximately 7% of eubacterial species, spanning diverse bacterial lineages. They seem to be significantly enriched in species that encode for more than one class 1 ASNase. Here, we report biochemical, biophysical, and structural properties of RrA, a member of scASNases family. Crystal structures of the wild-type RrA, both with and without bound L-Asp, as well as structures of several RrA mutants, reveal topologically unique tetramers. Moreover, the active site of one protomer is complemented by two residues (Tyr21 and Asn26) from another protomer. Upon closer inspection, these findings clearly outline scASNases as a stand-alone subfamily of ASNases that can catalyze the hydrolysis of L-Asn to L-Asp despite the lack of the C-terminal domain that is present in all ASNases described structurally to date.

Multifunctional Biocatalysts for Organic Synthesis.

Biocatalysis is becoming an indispensable tool in organic synthesis due to high enzymatic catalytic efficiency as well as exquisite chemo- and stereoselectivity. Some biocatalysts display great promiscuity including a broad substrate scope as well as the ability to catalyze more than one type of transformation. These promiscuous activities have been applied individually to efficiently access numerous valuable target molecules. However, systems in which enzymes possessing multiple different catalytic activities are applied in the synthesis are less well developed. Such multifunctional biocatalysts (MFBs) would simplify chemical synthesis by reducing the number of operational steps and enzyme count, as well as simplifying the sequence space that needs to be engineered to develop an efficient biocatalyst. In this Perspective, we highlight recently reported MFBs focusing on their synthetic utility and mechanism. We also offer insight into their origin as well as comment on potential strategies for their discovery and engineering.

Phylogeny-guided Characterization of Bacterial Hydrazine Biosynthesis Mediated by Cupin/methionyl tRNA Synthetase-like Enzymes.

Cupin/methionyl-tRNA synthetase (MetRS)-like didomain enzymes catalyze nitrogen-nitrogen (N-N) bond formation between Nω-hydroxylamines and amino acids to generate hydrazines, key biosynthetic intermediates of various natural products containing N-N bonds. While the combination of these two building blocks leads to the creation of diverse hydrazine products, the full extent of their structural diversity remains largely unknown. To explore this, we herein conducted phylogeny-guided genome-mining of related hydrazine biosynthetic pathways consisting of two enzymes: flavin-dependent Nω-hydroxylating monooxygenases (NMOs) that produce Nω-hydroxylamine precursors and cupin/MetRS-like enzymes that couple the Nω-hydroxylamines with amino acids via N-N bonds. A phylogenetic analysis identified the largely unexplored sequence spaces of these enzyme families. The biochemical characterization of NMOs demonstrated their capabilities to produce various Nω-hydroxylamines, including those previously not known as precursors of N-N bonds. Furthermore, the characterization of cupin/MetRS-like enzymes identified five new hydrazine products with novel combinations of building blocks, including one containing non-amino acid building blocks: 1,3-diaminopropane and putrescine. This study substantially expanded the variety of N-N bond forming pathways mediated by cupin/MetRS-like enzymes.

Narrow operators on [formula omitted]-complete lattice-normed spaces

We extend some of main results of Abasov et al., 2016, Fotiy et al., (2020), Mykhaylyuk et al., (2015), Pliev and Fang, (2017), Pliev and Popov, (2014) to the setting of orthogonally additive operators on lattice-normed spaces. We introduce a new class of C-complete lattice-normed spaces which strictly includes a class of Banach–Kantorovich spaces. The first main result of the paper asserts that every laterally-to-norm continuous C-compact orthogonally additive operator T:X→Y from an atomless C-complete lattice-normed space X to a Banach space Y is narrow. As a non expecting consequence of the first main result we obtain necessary and sufficient conditions for a nonlinear superposition operator TN:E(X)→E(X) to be C-compact. We also show that the sum of two orthogonally additive operators G and T, where G is narrow and T is laterally-to-norm continuous and C-compact, is a narrow operator. In the last part of the article we investigate dominated orthogonally additive operators. In particular we show the narrowness of dominated operators taking values in a Banach sequence space Y or in a separable symmetric operator space CE.

RNA compaction and iterative scanning for small RNA targets by the Hfq chaperone

RNA-guided enzymes must quickly search a vast sequence space for their targets. This search is aided by chaperones such as Hfq, a protein that mediates regulation by bacterial small RNAs (sRNAs). How RNA binding proteins enhance this search is little known. Using single-molecule Förster resonance energy transfer, we show that E. coli Hfq performs a one-dimensional scan in which compaction of the target RNA delivers sRNAs to sites distant from the location of Hfq recruitment. We also show that Hfq can transfer an sRNA between different target sites in a single mRNA, favoring the most stable duplex. We propose that compaction and segmental transfer, combined with repeated cycles of base pairing, enable the kinetic selection of optimal sRNA targets. Finally, we show that RNA compaction and sRNA transfer require conserved arginine patches. We suggest that arginine patches are a widespread strategy for enabling the movement of RNA across protein surfaces.

Machine learning to predict continuous protein properties from binary cell sorting data and map unseen sequence space

Proteins are a diverse class of biomolecules responsible for wide-ranging cellular functions, from catalyzing reactions to recognizing pathogens. The ability to evolve proteins rapidly and inexpensively toward improved properties is a common objective for protein engineers. Powerful high-throughput methods like fluorescent activated cell sorting and next-generation sequencing have dramatically improved directed evolution experiments. However, it is unclear how to best leverage these data to characterize protein fitness landscapes more completely and identify lead candidates. In this work, we develop a simple yet powerful framework to improve protein optimization by predicting continuous protein properties from simple directed evolution experiments using interpretable, linear machine learning models. Importantly, we find that these models, which use data from simple but imprecise experimental estimates of protein fitness, have predictive capabilities that approach more precise but expensive data. Evaluated across five diverse protein engineering tasks, continuous properties are consistently predicted from readily available deep sequencing data, demonstrating that protein fitness space can be reasonably well modeled by linear relationships among sequence mutations. To prospectively test the utility of this approach, we generated a library of stapled peptides and applied the framework to predict affinity and specificity from simple cell sorting data. We then coupled integer linear programming, a method to optimize protein fitness from linear weights, with mutation scores from machine learning to identify variants in unseen sequence space that have improved and co-optimal properties. This approach represents a versatile tool for improved analysis and identification of protein variants across many domains of protein engineering.

Effects and considerations of multiplexing ForenSeq Kintelligence libraries with a negative control.

The negative template control or negative amplification control has been an essential component of the forensic DNA analysis workflow that helps monitor contamination. As such, the inclusion of a negative control in forensic DNA analysis has been a requirement for all laboratories audited under the FBI's Quality Assurance Standards. As massively parallel sequencing (MPS) becomes more conventional in forensic laboratories, considerations for the inclusion of a negative control in every sequencing run can be evaluated. Although the inclusion of a negative control in library preparation and the first sequencing run has a practical function, there is less utility for its inclusion in all subsequent sequencing runs for that library preparation. Although this is universal to all MPS assays, it is most relevant for an assay that has a low sample multiplexing capacity, such as the ForenSeq Kintelligence Kit (Qiagen/Verogen, Inc.). The ForenSeq Kintelligence Kit is an investigative genetic genealogy (IGG) sequencing-based assay that targets 10,230 forensically relevant single-nucleotide polymorphisms. The manufacturer recommends multiplexing 3 libraries per sequencing run, which includes controls. The purpose of this study was to investigate the effect of the inclusion of a negative control in every Kintelligence sequencing run. We observed that the library generated from a negative amplification control will take 7%-14% of the run output. The loss of sequencing space taken by a negative control decreased the available output for DNA-containing samples, leading in some cases to allele or locus dropout and accompanying higher numbers of sixth to seventh order unknown associations in GEDmatch PRO.

New sequence spaces derived by using generalized arithmetic divisor sum function and compact operators

Abstract Define an infinite matrix D α = ( d n , v α ) \mathfrak{D}^{\alpha}=(d^{\alpha}_{n,v}) by d n , v α = { v α σ ( α ) ⁢ ( n ) , v ∣ n , 0 , v ∤ n , d^{\alpha}_{n,v}=\begin{cases}\dfrac{v^{\alpha}}{\sigma^{(\alpha)}(n)},&v\mid n,\\ 0,&v\nmid n,\end{cases} where σ ( α ) ⁢ ( n ) \sigma^{(\alpha)}(n) is defined to be the sum of the 𝛼-th power of the positive divisors of n ∈ N n\in\mathbb{N} , and construct the matrix domains ℓ p ⁢ ( D α ) \ell_{p}(\mathfrak{D}^{\alpha}) ( 0 < p < ∞ 0<p<\infty ), c 0 ⁢ ( D α ) c_{0}(\mathfrak{D}^{\alpha}) , c ⁢ ( D α ) c(\mathfrak{D}^{\alpha}) and ℓ ∞ ⁢ ( D α ) \ell_{\infty}(\mathfrak{D}^{\alpha}) defined by the matrix D α \mathfrak{D}^{\alpha} . We develop Schauder bases and determine 𝛼-, 𝛽- and 𝛾-duals of these new spaces. We characterize some matrix transformation from ℓ p ⁢ ( D α ) \ell_{p}(\mathfrak{D}^{\alpha}) , c 0 ⁢ ( D α ) c_{0}(\mathfrak{D}^{\alpha}) , c ⁢ ( D α ) c(\mathfrak{D}^{\alpha}) and ℓ ∞ ⁢ ( D α ) \ell_{\infty}(\mathfrak{D}^{\alpha}) to ℓ ∞ \ell_{\infty} , 𝑐, c 0 c_{0} and ℓ 1 \ell_{1} . Furthermore, we determine some criteria for compactness of an operator (or matrix) from X ∈ { ℓ p ⁢ ( D α ) , c 0 ⁢ ( D α ) , c ⁢ ( D α ) , ℓ ∞ ⁢ ( D α ) } X\in\{\ell_{p}(\mathfrak{D}^{\alpha}),c_{0}(\mathfrak{D}^{\alpha}),c(\mathfrak{D}^{\alpha}),\ell_{\infty}(\mathfrak{D}^{\alpha})\} to ℓ ∞ \ell_{\infty} , 𝑐, c 0 c_{0} or ℓ 1 \ell_{1} .

Mersenne Matrix Operator and Its Application in $p-$Summable Sequence Space

In this study, it is introduced the regular Mersenne matrix operator which is obtained by using Mersenne numbers and examined sequence spaces described as the domain of this matrix in the space of $p$-summable sequences for $1\leq p \leq \infty$. After that, it investigated some properties and inclusion relations, established the Schauder basis, and stated $\alpha-$, $\beta-$, and $\gamma-$duals of the aforementioned spaces. Additionally, it is characterized by the matrix classes from newly described spaces to classical sequence spaces. Finally, we studied the compactness of matrix operators on related sequence spaces.

Context-dependent design of induced-fit enzymes using deep learning generates well-expressed, thermally stable and active enzymes

The potential of engineered enzymes in industrial applications is often limited by their expression levels, thermal stability, and catalytic diversity. De novo enzyme design faces challenges due to the complexity of enzymatic catalysis. An alternative approach involves expanding natural enzyme capabilities for new substrates and parameters. Here, we introduce CoSaNN (Conformation Sampling using Neural Network), an enzyme design strategy using deep learning for structure prediction and sequence optimization. CoSaNN controls enzyme conformations to expand chemical space beyond simple mutagenesis. It employs a context-dependent approach for generating enzyme designs, considering non-linear relationships in sequence and structure space. We also developed SolvIT, a graph NN predicting protein solubility in Escherichia coli, optimizing enzyme expression selection from larger design sets. Using this method, we engineered enzymes with superior expression levels, with 54% expressed in E. coli, and increased thermal stability, with over 30% having higher Tm than the template, with no high-throughput screening. Our research underscores AI's transformative role in protein design, capturing high-order interactions and preserving allosteric mechanisms in extensively modified enzymes, and notably enhancing expression success rates. This method's ease of use and efficiency streamlines enzyme design, opening broad avenues for biotechnological applications and broadening field accessibility.

Extreme restructuring of cis-regulatory regions controlling a deeply conserved plant stem cell regulator.

A striking paradox is that genes with conserved protein sequence, function and expression pattern over deep time often exhibit extremely divergent cis-regulatory sequences. It remains unclear how such drastic cis-regulatory evolution across species allows preservation of gene function, and to what extent these differences influence how cis-regulatory variation arising within species impacts phenotypic change. Here, we investigated these questions using a plant stem cell regulator conserved in expression pattern and function over ~125 million years. Using in-vivo genome editing in two distantly related models, Arabidopsis thaliana (Arabidopsis) and Solanum lycopersicum (tomato), we generated over 70 deletion alleles in the upstream and downstream regions of the stem cell repressor gene CLAVATA3 (CLV3) and compared their individual and combined effects on a shared phenotype, the number of carpels that make fruits. We found that sequences upstream of tomato CLV3 are highly sensitive to even small perturbations compared to its downstream region. In contrast, Arabidopsis CLV3 function is tolerant to severe disruptions both upstream and downstream of the coding sequence. Combining upstream and downstream deletions also revealed a different regulatory outcome. Whereas phenotypic enhancement from adding downstream mutations was predominantly weak and additive in tomato, mutating both regions of Arabidopsis CLV3 caused substantial and synergistic effects, demonstrating distinct distribution and redundancy of functional cis-regulatory sequences. Our results demonstrate remarkable malleability in cis-regulatory structural organization of a deeply conserved plant stem cell regulator and suggest that major reconfiguration of cis-regulatory sequence space is a common yet cryptic evolutionary force altering genotype-to-phenotype relationships from regulatory variation in conserved genes. Finally, our findings underscore the need for lineage-specific dissection of the spatial architecture of cis-regulation to effectively engineer trait variation from conserved productivity genes in crops.

On solutions vanishing at infinity of infinite systems of quadratic Urysohn integral equations

The paper is devoted to present a result on the existence of solutions of an infinite system of quadratic integral equations of the Urysohn type considered on the real half-axis. Our investigations are conducted in the Banach space consisting of bounded and continuous functions defined on the real half-axis with values in the space of real sequences converging to zero. That space is equipped with the standard supremum norm. The main tools used in our study is the technique of measures of noncompactness and the Schauder fixed point principle. We illustrate our result by a suitable example.

Nanopatterned Monolayers of Bioinspired, Sequence-Defined Polypeptoid Brushes for Semiconductor/Bio Interfaces.

The ability to control and manipulate semiconductor/bio interfaces is essential to enable biological nanofabrication pathways and bioelectronic devices. Traditional surface functionalization methods, such as self-assembled monolayers (SAMs), provide limited customization for these interfaces. Polymer brushes offer a wider range of chemistries, but choices that maintain compatibility with both lithographic patterning and biological systems are scarce. Here, we developed a class of bioinspired, sequence-defined polymers, i.e., polypeptoids, as tailored polymer brushes for surface modification of semiconductor substrates. Polypeptoids featuring a terminal hydroxyl (-OH) group are designed and synthesized for efficient melt grafting onto the native oxide layer of Si substrates, forming ultrathin (∼1 nm) monolayers. By programming monomer chemistry, our polypeptoid brush platform offers versatile surface modification, including adjustments to surface energy, passivation, preferential biomolecule attachment, and specific biomolecule binding. Importantly, the polypeptoid brush monolayers remain compatible with electron-beam lithographic patterning and retain their chemical characteristics even under harsh lithographic conditions. Electron-beam lithography is used over polypeptoid brushes to generate highly precise, binary nanoscale patterns with localized functionality for the selective immobilization (or passivation) of biomacromolecules, such as DNA origami or streptavidin, onto addressable arrays. This surface modification strategy with bioinspired, sequence-defined polypeptoid brushes enables monomer-level control over surface properties with a large parameter space of monomer chemistry and sequence and therefore is a highly versatile platform to precisely engineer semiconductor/bio interfaces for bioelectronics applications.

Exponential Combination of a and e/g Intracellular Peptide Libraries Identifies a Selective ATF3 Inhibitor.

Activating transcription factor 3 (ATF3) is an activation transcription factor/cyclic adenosine monophosphate (cAMP) responsive element-binding (CREB) protein family member. It is recognized as an important regulator of cancer progression by repressing expression of key inflammatory factors such as interferon-γ and chemokine (C-C motif) ligand 4 (CCL4). Here, we describe a novel library screening approach that probes individual leucine zipper components before combining them to search exponentially larger sequence spaces not normally accessible to intracellular screening. To do so, we employ two individual semirational library design approaches and screen using a protein-fragment complementation assay (PCA). First, a 248,832-member library explored 12 amino acid positions at all five a positions to identify those that provided improved binding, with all e/g positions fixed as Q, placing selection pressure onto the library options provided. Next, a 59,049-member library probed all ten e/g positions with 3 options. Similarly, during e/g library screening, a positions were locked into a generically bindable sequence pattern (AIAIA), weakly favoring leucine zipper formation, while placing selection pressure onto e/g options provided. The combined a/e/g library represents ∼14.7 billion members, with the resulting peptide, ATF3W_aeg, binding ATF3 with high affinity (Tm = 60 °C; Kd = 151 nM) while strongly disfavoring homodimerization. Moreover, ATF3W_aeg is notably improved over component PCA hits, with target specificity found to be driven predominantly by electrostatic interactions. The combined a/e/g exponential library screening approach provides a robust, accelerated platform for exploring larger peptide libraries, toward derivation of potent yet selective antagonists that avoid homoassociation to provide new insight into rational peptide design.

On I-Convergence Difference Sequence Spaces Defined by Orlicz Function in 2−Normed Space

The difference sequence spaces of type I-convergent, I-null, bounded I-convergent, and bounded I-null in 2−normed space are introduced and studied using the Orlicz function. Investigations into the pertinent characteristics of these spaces led us to the establishment of some inclusion relations.

On Topological Structure of Total Paranormed Double Sequence Space (ℓ 2 ((X, ||.||), γ, ¯ w¯), G)

The aim of this paper is to introduce and study a new class ℓ2 ((X, ||.||), γ̅, ω̅ ) of double sequences with their terms in a normed space X as a generalization of the familiar sequence space ℓ. Besides the investigation of the condition pertaining to the containment relations of the class ℓ2 ((X, ||.||), γ̅, ω̅ ) of same kind in terms of γ̅ and ω̅, our primary interest is to explore some of the preliminary results that characterize the linear topological structures of ℓ2 ((X, ||.||), γ̅, ω̅ ) when topologized it with suitable natural paranorm.

Predicting S. aureus antimicrobial resistance with interpretable genomic space maps.

Increasing antimicrobial resistance (AMR) represents a global healthcare threat. To decrease the spread of AMR and associated mortality, methods for rapid selection of optimal antibiotic treatment are urgently needed. Machine learning (ML) models based on genomic data to predict resistant phenotypes can serve as a fast screening tool prior to phenotypic testing. Nonetheless, many existing ML methods lack interpretability. Therefore, we present a methodology for visualization of sequence space and AMR prediction based on the non-linear dimensionality reduction method - generative topographic mapping (GTM). This approach, applied to AMR data of >5000 S. aureus isolates retrieved from the PATRIC database, yielded GTM models with reasonable accuracy for all drugs (balanced accuracy values ≥0.75). The Generative Topographic Maps (GTMs) represent data in the form of illustrative maps of the genomic space and allow for antibiotic-wise comparison of resistant phenotypes. The maps were also found to be useful for the analysis of genetic determinants responsible for drug resistance. Overall, the GTM-based methodology is a useful tool for both the illustrative exploration of the genomic sequence space and AMR prediction.

Mapping Composition Evolution through Synthesis, Purification, and Depolymerization of Random Heteropolymers.

Random heteropolymers (RHPs) consisting of three or more comonomers have been routinely used to synthesize functional materials. While increasing the monomer variety diversifies the side-chain chemistry, this substantially expands the sequence space and leads to ensemble-level sequence heterogeneity. Most studies have relied on monomer composition and simulated sequences to design RHPs, but the questions remain unanswered regarding heterogeneities within each RHP ensemble and how closely these simulated sequences reflect the experimental outcomes. Here, we quantitatively mapped out the evolution of monomer compositions in four-monomer-based RHPs throughout a design-synthesis-purification-depolymerization process. By adopting a Jaacks method, we first determined 12 reactivity ratios directly from quaternary methacrylate RAFT copolymerization experiments to account for the influences of competitive monomer addition and the reversible activation/deactivation equilibria. The reliability of in silico analysis was affirmed by a quantitative agreement (<4% difference) between the simulated RHP compositions and the experimental results. Furthermore, we mapped out the conformation distribution within each ensemble in different solvents as a function of monomer chemistry, composition, and segmental characteristics via high-throughput computation based on self-consistent field theory (SCFT). These comprehensive studies confirmed monomer composition as a viable design parameter to engineer RHP-based functional materials as long as the reactivity ratios are accurately determined and the livingness of RHP synthesis is ensured.

A generalization of Hardy’s inequality to infinite tensors

Abstract In this paper, we extend Hardy’s inequality to infinite tensors. To do so, we introduce Cesàro tensors ℭ {\mathfrak{C}} , and consider them as tensor maps from sequence spaces into tensor spaces. In fact, we prove inequalities of the form ∥ ℭ ⁢ x k ∥ t , 1 ≤ U ⁢ ∥ x ∥ l p k \|\mathfrak{C}x^{k}\|_{t,1}\leq U\|x\|_{l_{p}}^{k} ( k = 1 , 2 k=1,2 ), where x is a sequence, ℭ ⁢ x k {\mathfrak{C}x^{k}} is a tensor, and ∥ ⋅ ∥ t , 1 {\|\cdot\|_{t,1}} , ∥ ⋅ ∥ l p {\|\cdot\|_{l_{p}}} are the tensor and sequence norms, respectively. The constant U is independent of x, and we seek the smallest possible value of U.