Article pubs.acs.org/journal/abseba Matriarch: A Python Library for Materials Architecture † ‡ † ‡ § † ⊥ ‡ Tristan Giesa, , Ravi Jagadeesan, , , David I. Spivak, , and Markus J. Buehler*, ‡ ⊥ Laboratory for Atomistic and Molecular Mechanics, Department of Civil and Environmental Engineering, and Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, United States § Harvard University, 1 Oxford Street, Cambridge, Massachusetts 02138, United States *S Supporting Information ABSTRACT: Biological materials, such as proteins, often have a hierarchical structure ranging from basic building blocks at the nanoscale (e.g., amino acids) to assembled structures at the macroscale (e.g., fibers). Current software for materials engineering allows the user to specify polypeptide chains and simple secondary structures prior to molecular dynamics simulation, but is not flexible in terms of the geometric arrange- ment of unequilibrated structures. Given some knowledge of a larger-scale structure, instructing the software to create it can be very difficult and time-intensive. To this end, the present paper reports a mathematical language, using category theory, to describe the architecture of a material, i.e., its set of building blocks and instructions for combining them. While this framework applies to any hierarchical material, here we concentrate on proteins. We implement this mathematical language as an open-source Python library called Matriarch. It is a domain-specific language that gives the user the ability to create almost arbitrary structures with arbitrary amino acid sequences and, from them, generate Protein Data Bank (PDB) files. In this way, Matriarch is more powerful than commercial software now available. Matriarch can be used in tandem with molecular dynamics simulations and helps engineers design and modify biologically inspired materials based on their desired functionality. As a case study, we use our software to alter both building blocks and building instructions for tropocollagen, and determine their effect on its structure and mechanical properties. KEYWORDS: hierarchical protein materials, building block, structure creation, category theory, molecular design, open-source software 1. INTRODUCTION Specifically, during the creation of the initial structure for a Despite the increasing interest in using hierarchically structured molecular dynamics simulation, it may be useful to directly control both the sequence and the building instructions, such protein materials as low-cost, environmentally benign, and yet fi high-performance materials, the interplay between nanoscale as the pitch of a helix or the gap distance of a bril composite. structure and global properties is not well understood. Recent Current software does not allow one to parametrically change ’ such building instructions. To this end, we develop a domain- advances in algorithms for predicting a protein s structure from fi its amino acid sequence1 have facilitated the engineering of speci c language of materials architecture and implement it in Publication Date (Web): September 1, 2015 | doi: 10.1021/acsbiomaterials.5b00251 2−4 an open-source Python library, called Matriarch. This program, new catalysts and other proteins with new structures and fi functions.5 Biologically inspired engineering6 and materiomic which is based on the mathematical eld of category theory, is engineering7,8 seek to build multiscale hierarchical materials discussed in section 2. Although the mathematics applies to a wide variety of materials, Matriarch is currently restricted to Downloaded by MASSACHUSETTS INST OF TECHNOLOGY on September 11, 2015 | http://pubs.acs.org based on their desired functionality, and have the potential to generate many new materials on demand.9 proteins architecture. Materials engineers explore the design-space for sequence, Matriarch approximates structures of proteins and generates fi structure, and functionality of organic materials.10,11 In this con- atomic con gurations from a given material architecture. Having fi text, the building block replacement problem12 poses the question a reasonable initial con guration, produced by Matriarch in the fi fi of how global structure and function of a molecule is affected by form of a PDB le, can signi cantly shorten molecular dynamics basic building block substitutions, and furthermore how to use equilibration times and associated structure and function analysis. these substitutions to achieve desired properties. For example, in The engineer can then perform building block substitutions and the case of proteins, one can consider the effect of replacing amino variations on building instructions, using Matriarch, to study ff acids in a polypeptide by other amino acids. The building instruc- their e ect on the functionality of a material. Matriarch thus tions, by which these building blocks are recursively put together facilitates the design process for the materials engineer (see into a structure for self-assembly, are an equally important part of Figure 1). the process. We refer to the combination of building blocks and associated building instructions as materials architecture. Received: June 10, 2015 Control over such architecture is of interest when there is an Accepted: August 11, 2015 incomplete knowledge of a protein’s final assembled structure. © XXXX American Chemical Society A DOI: 10.1021/acsbiomaterials.5b00251 ACS Biomater. Sci. Eng. XXXX, XXX, XXX−XXX ACS Biomaterials Science & Engineering Article Figure 1. Role of materials architecture in the engineering process. A material architecture consists of building blocks arranged according to building instructions, formalized using category theory. The building blocks of proteins are amino acids, which can be arranged in any number of forms by building instructions. Matriarch is a Python implementation of the mathematical formalism, which realizes the chosen materials architecture as a PDB file. The structure and function of the prearranged molecules can be obtained by molecular dynamics simulation. Using the data from many such simulations informs the design process for the materials engineer. Section 3.2 provides a proof of concept for the utility of changing both building blocks and building instructions. We first develop a building instruction for forming tropocollagen-shaped molecules (triple-helices). We then systematically substitute parts of the tropocollagen polypeptide sequence, and shape the mutated polypeptide into a triple helix Figure 2. Architecture of rectangle arrangements. Equivalent to an early using our building instruction. Through equilibration and example of an operad from mathematics called the little 2-cubes operad steered molecular dynamics simulations, we observe a large E . In the terminology of this paper, the only building blocks in E are variation in the elastic moduli of the tropocollagen variants. 2 2 rectangles. A building instruction in E2 is an arrangement of nonoverlapping Using Matriarch, similar methods can be extended to other rectangles within a larger rectangle. In the composition, the placements and building block replacement questions. Also in section 3,we aspect ratios of building blocks are retained, but sizes can change. The fi parametrically create structures, such as a triangular helix of an operad E2 is the mathematical description of rectangle con gurations as amyloid, which would be extremely difficult and time-intensive they are nested into hierarchies of arbitrary depth. The operad underlying to create using existing software. Matriarch is far more complex than E2, with more diverse building blocks and building instructions. 2. THEORETICAL FRAMEWORK The language of materials architecture is formalized using a Our operad M has protein building block types as objects, concept within category theory called operads, which can model and protein building instructions as morphisms. Forming a Publication Date (Web): September 1, 2015 | doi: 10.1021/acsbiomaterials.5b00251 − the assembly of hierarchical structures.13 15 Figure 2 provides a sequence of instructions, or program, in Matriarch corresponds classical example from mathematics, in which the only building to composing a tree of morphisms in the operad. A set-valued blocks are rectangles, and the building instructions are placements functor M → Set assigns to each object (building block type) Downloaded by MASSACHUSETTS INST OF TECHNOLOGY on September 11, 2015 | http://pubs.acs.org of nonoverlapping rectangles in a larger rectangle. The operad is the set of all building blocks of this type, and it assigns to each the mathematical description of rectangle configurationsasthey morphism (building instruction) a function that creates a new are nested into hierarchies of arbitrary depth. building block by combining or manipulating existing building In the operad describing materials architecture, specifically blocks. The detailed mathematical framework is developed for that of hierarchical protein materials, both the building blocks experts in the Mathematics Supplement. A more practical and the building instructions are far more complex. It is useful formulation can be found in a User’s Guide,16 and a summary is to have such a mathematical language because it deals only with given in sections 2.1−2.2 below. abstract building blocks and avoids the specifics of molecular 2.1. Building Blocks. A building block is abstracted to a geometry. That is, the abstractions offer a unified language set of oriented rigid bodies