Stuart M. Marshall’s research while affiliated with University of Glasgow and other places

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Publications (14)


Experimentally measured assembly indices are required to determine the threshold for life
  • Article
  • Full-text available

November 2024

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52 Reads

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1 Citation

Sara I Walker

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Stuart Marshall

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Leroy Cronin

Assembly theory (AT) aims to distinguish living from non-living systems by explaining and quantifying selection and evolution. The theory proposes that the degree of assembly depends on the number of complex objects, with complexity measured using a combination of the object’s assembly index (AI) and its abundance. We previously provided experimental evidence supporting AT’s predictive power, finding that abiotic systems do not randomly produce organic molecules with an AI greater than approximately 15 in detectable amounts. Hazen et al. (Hazen et al. 2024 J. R. Soc. Interface 21, 20230632. (doi:10.1098/rsif.2023.0632)) proposed inorganic molecules that theoretically have AIs greater than 15, suggesting similar complexity to biological molecules. However, our AIs are experimentally measured for organic, covalently bonded molecules, whereas Hazen’s are theoretical, derived from crystal structures of charged units that are not isolable in solution. This distinction underscores the challenge in experimentally validating theoretical AIs.

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Experimental Measurement of Assembly Indices are Required to Determine The Threshold for Life

June 2024

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126 Reads

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1 Citation

Assembly Theory (AT) was developed to help distinguish living from non-living systems. The theory is simple as it posits that the amount of selection or Assembly is a function of the number of complex objects where their complexity can be objectively determined using assembly indices. The assembly index of a given object relates to the number of recursive joining operations required to build that object and can be not only rigorously defined mathematically but can be experimentally measured. In pervious work we outlined the theoretical basis, but also extensive experimental measurements that demonstrated the predictive power of AT. These measurements showed that is a threshold in assembly indices for organic molecules whereby abiotic chemical systems could not randomly produce molecules with an assembly index greater or equal than 15. In a recent paper by Hazen et al [1] the authors not only confused the concept of AT with the algorithms used to calculate assembly indices, but also attempted to falsify AT by calculating theoretical assembly indices for objects made from inorganic building blocks. A fundamental misunderstanding made by the authors is that the threshold is a requirement of the theory, rather than experimental observation. This means that exploration of inorganic assembly indices similarly requires an experimental observation, correlated with the theoretical calculations. Then and only then can the exploration of complex inorganic molecules be done using AT and the threshold for living systems, as expressed with such building blocks, be determined. Since Hazen et al.[1] present no experimental measurements of assembly theory, their analysis is not falsifiable.


Investigating and Quantifying Molecular Complexity Using Assembly Theory and Spectroscopy

April 2024

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50 Reads

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23 Citations

ACS Central Science

Michael Jirasek

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Abhishek Sharma

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Jessica R. Bame

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[...]

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Leroy Cronin

Current approaches to evaluate molecular complexity use algorithmic complexity, rooted in computer science, and thus are not experimentally measurable. Directly evaluating molecular complexity could be used to study directed vs undirected processes in the creation of molecules, with potential applications in drug discovery, the origin of life, and artificial life. Assembly theory has been developed to quantify the complexity of a molecule by finding the shortest path to construct the molecule from building blocks, revealing its molecular assembly index (MA). In this study, we present an approach to rapidly infer the MA of molecules from spectroscopic measurements. We demonstrate that the MA can be experimentally measured by using three independent techniques: nuclear magnetic resonance (NMR), tandem mass spectrometry (MS/MS), and infrared spectroscopy (IR). By identifying and analyzing the number of absorbances in IR spectra, carbon resonances in NMR, or molecular fragments in tandem MS, the MA of an unknown molecule can be reliably estimated. This represents the first experimentally quantifiable approach to determining molecular assembly. This paves the way to use experimental techniques to explore the evolution of complex molecules as well as a unique marker of where an evolutionary process has been operating.


Fig 1. Molecular Assembly of 5-aminoisophathalic acid. (A) Molecular Assembly pathway of 5-aminoisophathalic with a total of 7 steps. The various chemical bonds are considered as fundamental building blocks (shown in red) and the substructures (shown in blue) along the pathways constitute the assembly pool. (B-D) Experimental NMR, IR, and MS 2 spectra of 5-aminoisophathalic acid highlight different features of the molecule from which the molecular constraints and the MA can be inferred.
Fig. 3. Inferring Molecular Assembly from infrared spectroscopy. (A) XTB-calculated IR spectrum of 5-aminoisophthalic acid with highlighted fingerprint region (400-1500 cm -1 ). (B) Example of the six most intense vibrational bands in the fingerprint region demonstrating its collective-motion nature. (C) Assembly index vs. IR-inferred assembly index estimated from the number of IR peaks in the fingerprint region (400-1500 cm -1 ) based on XTB calculation on 10,000 molecules (see Eq. 1). Correlation between the predicted and expected assembly index is 0.86. (D) Assembly index vs. IR-inferred assembly index estimated from the number of IR peaks in the fingerprint region (400-1500 cm -1 ) based on the experimental measurement on 99 molecules (see Eq. 2). Correlation between the predicted and expected assembly index is 0.75.
Fig. 4. Inferring molecular complexity from 13 C NMR spectra. (A) and (B) shows the predicted 13 C NMR spectrum of 5-aminoisophathalic acid and quinine, with highlighted different types of carbons. (C) Assembly index vs. NMR-inferred assembly index estimated from the number of different types of carbons (see Eq. 3) based on NMRshiftDB calculation on 10,000 molecules. The correlation between the predicted and expected assembly index is 0.87. (D) Assembly index vs. NMR-inferred assembly index estimated from the number of different types of carbons experimentally on 101 molecules, using the same model as in the theoretical set. The correlation between the predicted and expected assembly index is 0.81.
Fig. 5. Example of deconvolution of mixture complexity. (A) Overlay of spectra using DEPTQ-90 and DEPTQ-135 methods to differentiate different types of carbons. (B) 13 C-DOSY deconvolution of peaks to the individual compound assignment via diffusion. (C) Assigned deconvolution of 13 C-NMR spectra, with peaks for 5-aminoisophathalic acid in red and quinine in blue.
Fig. 6 LC-MS Mixture Analysis allows the determination of MA number for individual molecules. (A) Ordinary Linear Regression between calculated MA and average MS 2 peaks (from previous work), including small molecules and peptides. (B) Calculated and predicted MA using the linear regression model on the same dataset with a Pearson Correlation Coefficient of 0.84. (C) Chromatogram of a mixture of ten molecules with seven molecules identified. (D-F) MS 2 spectra of Ceftiofur, Succinylsulfathiazole, and Sildenafil respectively showing compounds from the mixture can be resolved.

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Multimodal Techniques for Detecting Alien Life using Assembly Theory and Spectroscopy

February 2023

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250 Reads

Detecting alien life is a difficult task because it's hard to find signs of life that could apply to any life form. However, complex molecules could be a promising indicator of life and evolution. Currently, it's not possible to experimentally determine how complex a molecule is and how that correlates with information-theoretic approaches that estimate molecular complexity. Assembly Theory has been developed to quantify the complexity of a molecule by finding the shortest path to construct the molecule from simple parts, revealing its molecular assembly index (MA). In this study, we present an approach to rapidly and exhaustively calculate molecular assembly and explore the MA of over 10,000 molecules. We demonstrate that molecular complexity (MA) can be experimentally measured using three independent techniques: nuclear magnetic resonance (NMR), tandem mass spectrometry (MS), and infrared spectroscopy (IR), and these give consistent results with good correlations. By identifying and counting the number of absorbances in IR spectra, carbon resonances in NMR, or molecular fragments in tandem MS, the molecular assembly index of an unknown molecule can be reliably estimated from experimental data. This represents the first experimentally quantifiable approach to defining molecular assembly, a reliable metric for complexity, as an intrinsic property of all molecules and can also be performed on complex mixtures. This paves the way to use spectroscopic techniques to unambiguously detect alien life in the solar system, and beyond on exoplanets.



Figure 2. The basic assembly concept is demonstrated here. Each of the final structures can be created from white and blue basic objects in four joining operations, giving an assembly index of 4. Pathway (a) shows the creation of a structure that can only be formed in four steps by adding one basic object at a time, while pathway (c) represents the maximum increase in size per step, by combining the largest object in the pathway with itself at each stage. Pathway (b) is an intermediate case.
Figure 4. An assembly map that maps an assembly space of white and blue blocks onto integers representing the object size.
Figure 5. Example assembly pathways for systems of varying dimensionality.
Figure 7. Illustrative assembly pathway of a two-dimensional image. This does not necessarily represent the minimal assembly pathway for this shape. Here, images that are rotated or reflected are considered equivalent.
Formalising the Pathways to Life Using Assembly Spaces

June 2022

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187 Reads

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55 Citations

Entropy

Assembly theory (referred to in prior works as pathway assembly) has been developed to explore the extrinsic information required to distinguish a given object from a random ensemble. In prior work, we explored the key concepts relating to deconstructing an object into its irreducible parts and then evaluating the minimum number of steps required to rebuild it, allowing for the reuse of constructed sub-objects. We have also explored the application of this approach to molecules, as molecular assembly, and how molecular assembly can be inferred experimentally and used for life detection. In this article, we formalise the core assembly concepts mathematically in terms of assembly spaces and related concepts and determine bounds on the assembly index. We explore examples of constructing assembly spaces for mathematical and physical objects and propose that objects with a high assembly index can be uniquely identified as those that must have been produced using directed biological or technological processes rather than purely random processes, thereby defining a new scale of aliveness. We think this approach is needed to help identify the new physical and chemical laws needed to understand what life is, by quantifying what life does.


Fig. 1. Representations of an assembly pathway, by taking adenine as an example. (A) One of the many assembly pathways of adenine (it turns out to be the shortest one, according to our MC algorithm, explained later). The assembly pool (shown inside the dashed boxes) evolves with each assembly step. The colors denote which two assembly building blocks are used to make the new one (note that the color schemes are independent for each step). (B) The key-step representation of the assembly pathway. (C) The joint process for each key assembly step, which is used to work out the multiset representation. (D) The multiset representation of this assembly pathway. Strictly speaking, it should be written as {[1] 1 , [2] 1 } where the superscript "1" is the multiplicity of this assembly building block, that is, after canceling out, it appears once on the left-hand side of (C), but for simplicity, we only explicitly write down the multiplicity when it is not 1.
Fig. 2. Two exemplified molecular assembly trees. (A) The assembly space of adenine and thymine. The shortest assembly pathway for adenine alone is indicated by the blue dashed arrows, while the shortest assembly pathway for thymine alone is indicated by the red dashed arrows. The shortest assembly pathway to make adenine and thymine altogether is the one indicated by the black dashed arrows. (B) A molecular assembly tree for A, G, T, U, and C, which can also be written as {[2, 10, 11, 12, 13]}, whose index is calculated to be 16. Note that, in both (A) and (B), the colors are just used to help the reader recognize the building blocks, and the color schemes are independent; we also omitted the arrows starting from the basic building blocks for a better visualization.
Fig. 5. The assembly tree of 10 commonly used plasticizers including BBP, DEHP, DEHA, and others. For a clearer visualization, all plasticizers are made dimmer than other parts of the tree. The most central structures are highlighted green.
Fig. 6. The assembly tree of nine compounds in the family of opiates and one -opioid receptor agonist (salvinorin A). Some of these opiates are natural (morphine, codeine, thebaine, and papaverine), while others are synthetic (fentanyl, remifentanil, methadone, pethidine, and diamorphine, also known as heroin). For a clearer visualization, all opioids are made dimmer than other parts of the tree.
Fig. 7. Comparison between natural opiates and opiate-like molecules generated using Reassembler. (A) shows the six opiates used to generate the assembly pools, and (B) shows six new opiate-like molecules generated from those assembly pools. See section S7.3 for more detailed information on more new compounds.
Exploring and mapping chemical space with molecular assembly trees

September 2021

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158 Reads

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63 Citations

Science Advances

The rule-based search of chemical space can generate an almost infinite number of molecules, but exploration of known molecules as a function of the minimum number of steps needed to build up the target graphs promises to uncover new motifs and transformations. Assembly theory is an approach to compare the intrinsic complexity and properties of molecules by the minimum number of steps needed to build up the target graphs. Here, we apply this approach to prebiotic chemistry, gene sequences, plasticizers, and opiates. This allows us to explore molecules connected to the assembly tree, rather than the entire space of molecules possible. Last, by developing a reassembly method, based on assembly trees, we found that in the case of the opiates, a new set of drug candidates could be generated that would not be accessible via conventional fragment-based drug design, thereby demonstrating how this approach might find application in drug discovery.


Fig. 2 Molecular assembly and chemical space. a Schematic of assembly paths for four example molecules (hydrogens and charges omitted for clarity). b The computed MA of molecules from the Reaxys database shown by molecular weight. The color scale indicates the frequency, with increasing frequency from dark purple (0.0) to green and yellow (1.0) of molecules in a given molecular weight range with a given MA. 2.5 million MA were calculated, in the figure shown here that data has been subsampled to control for bias, see SI. The overlaid plot with the white labels shows how the MA varies for some compound types where some natural products, pharmaceuticals, and metabolites have a wide range of values (these molecules are listed in Supplementary Information Section 7, table 2). Note that the range of MA for the amino acids is limited. The molecular masses are binned in 50 Dalton sections. c Example organic molecular structures and the corresponding MA values calculated.
Assembly pathways
a In analyzing the assembly pathways of an object, we start with its basic building blocks, which are the shared set of objects that can construct our target object and any other object within the class of objects. The Assembly index of an object is defined as the smallest number of joining operations required to create the object using this model. b We can model the assembly process as a random walk on weighted trees where the number of outgoing edges (leaves) grows as a function of the depth of the tree, due to the addition of previously made sub-structures. By generating several million trees and calculating the likelihood of the most likely path through the tree, we can estimate the likelihood of an object forming by chance as a function of the number of joining operations required (path length). c The probability of the most likely path through the tree as a function of the path length decreases rapidly. The colors indicate different assumptions about the chemical space. For comparison, the dashed lines indicate the ratio of (I) one star in the entire milky way, 1:10¹¹, (II) one gram out of all of Earth’s biomass, 1:10¹⁷, (III) one in a mole, 1:10²³, and (IV) one gram out of Earth’s mass (1:10²⁹). Note on this plot the path probability of the formation of Taxol would vary between 1:10³⁵ to 1:10⁶⁰ with a path length of 30 and the amount of chemical predisposition is varied with alpha biasing the effective selectivity between 50–99.9% at each step respectively.
Experimental correlation of mass spectrometry data to MA and MA analysis of mixtures
a Three example molecular structures with associated MA index. b The fragmentation spectra associated with the molecular ions from (A). The high MA molecules have more peaks in their fragmentation spectra. c The observed correlation between the number of peaks in a fragmentation spectrum and the MA value of the ion, the shaded region shows the 90% prediction interval using quantile regression, with the median prediction shown in the center line. The circles represent small organics while triangles represent peptides. D-F indicate analytical workflow for measuring MA in mixtures. d A single ion is selected based on intensity. e MS2 spectra from the selected ion, with the inset showing the same spectra zoomed in on the shaded region to show lower intensity peaks. The total number of peaks in the fragmentation spectra are counted to correlate with the MA. f Many ions from the mixture will be fragmented and the predicted MA from that sample form a distribution, we consider the highest MA value measured to represent the MA of the mixture.
Estimated MA of laboratory and environmental samples
a The estimated MA against the parent mass of many ions for different samples in the 300–500 m/z range (excluding Taxol with has a m/z value of 854.9). b The distributions of estimated MA for all samples split by category, colored by source, the inset shows the distribution of points for a single biological sample, E. Coli. The MA of biological samples has a wider distribution, showing that only biologically produced samples produce MA above a certain threshold. c The estimated MA values for each sample with the blinded identities correctly labeled. The highest MA value in each sample is bold and the lower values faded. Each sample may have more than 15 points due to the dynamic exclusion settings used, which enable us to collect more MS2 peaks. Samples may have less than 15 points due to excluding noisy or unreliable spectra, for more information see Supplementary Information Section 5. *These samples were run with a column attached to the mass spectrometry but no chromatographic method was used. °This sample was gathered from an online database and analyzed with a different instrument. † Taxol is shown in Fig. 4C but has a mass that is not shown in Fig. 4A or 5B. See Supplementary Information Section 6 for details.
Identifying molecules as biosignatures with assembly theory and mass spectrometry

May 2021

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625 Reads

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153 Citations

The search for alien life is hard because we do not know what signatures are unique to life. We show why complex molecules found in high abundance are universal biosignatures and demonstrate the first intrinsic experimentally tractable measure of molecular complexity, called the molecular assembly index (MA). To do this we calculate the complexity of several million molecules and validate that their complexity can be experimentally determined by mass spectrometry. This approach allows us to identify molecular biosignatures from a set of diverse samples from around the world, outer space, and the laboratory, demonstrating it is possible to build a life detection experiment based on MA that could be deployed to extraterrestrial locations, and used as a complexity scale to quantify constraints needed to direct prebiotically plausible processes in the laboratory. Such an approach is vital for finding life elsewhere in the universe or creating de-novo life in the lab. The search for life in the universe is difficult due to issues with defining signatures of living systems. Here, the authors present an approach based on the molecular assembly number and tandem mass spectrometry that allows identification of molecules produced by biological systems, and use it to identify biosignatures from a range of samples, including ones from outer space.


Mapping the Molecular Tree of Life using Assembly Spaces

December 2020

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105 Reads

p> The mathematical search of chemical space can generate an almost infinite number of molecules and it is hard to know which molecules are relevant experimentally. A way to explore the chemical space of known molecules as a function of their relative complexity might help us understand biological processes and find new relationships. Assembly theory provides an approach to explore and compare the intrinsic complexity of molecules by the minimum number of steps needed to build up the target graphs. Here we show assembly theory can be applied to networks of molecules to explore the assembly properties of common motifs and use these to define a tree of assembly spaces. This theory allows us to explore the accessible molecules connected to the tree, rather than the entire space of possible molecules. We apply this approach to prebiotic chemistry, to gene sequences, a family of plasticizers, as well as the well-known opiate class of natural products. This analysis allows us to quantify the amount of external information needed to assemble the tree and identify and predict new components in this family of molecules. Finally, by developing a new reassembly system that uses the disassembly motifs, we found that in the case of the opiates a new set of opiate-like drug candidates could be generated that would not be accessible via conventional fragment-based drug design, thereby demonstrating how this approach might find application in drug discovery. </p


Identifying Molecules as Biosignatures with Assembly Theory and Mass Spectrometry

November 2020

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68 Reads

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2 Citations

p> The search for evidence of life elsewhere in the universe is hard because it is not obvious what signatures are unique to life. Here we postulate that complex molecules found in high abundance are universal biosignatures as they cannot form by chance. To explore this, we developed the first intrinsic measure of molecular complexity that can be experimentally determined, and this is based upon a new approach called assembly theory which gives the molecular assembly number (MA) of a given molecule. MA allows us to compare the intrinsic complexity of molecules using the minimum number of steps required to construct the molecular graph starting from basic objects, and a probabilistic model shows how the probability of any given molecule forming randomly drops dramatically as its MA increases. To map chemical space, we calculated the MA of ca. 2.5 million compounds, and collected data which showed the complexity of a molecule can be experimentally determined by using three independent techniques including infra-red spectroscopy, nuclear magnetic resonance, and by fragmentation in a mass spectrometer, and this data has an excellent corelation with the values predicted from our assembly theory. We then set out to see if this approach could allow us to identify molecular biosignatures with a set of diverse samples from around the world, outer space, and the laboratory including prebiotic soups. The results show that there is a non-living to living threshold in MA complexity and the higher the MA for a given molecule, the more likely that it had to be produced by a biological process . This work demonstrates it is possible to use this approach to build a life detection instrument that could be deployed on missions to extra-terrestrial locations to detect biosignatures, map the extent of life on Earth, and be used as a molecular complexity scale to quantify the constraints needed to direct prebiotically plausible processes in the laboratory. Such an approach is vital if we are going to find new life elsewhere in the universe or create de-novo life in the lab. </p


Citations (10)


... We clarify misunderstandings of Walker et al. (Walker et al. 2024 J. R. Soc. Interface 21, 20240367 (doi:10.1098/rsif.2024.0367)) ...

Reference:

Reply to ‘Experimental measurement of assembly indices are required to determine the threshold for life’
Experimentally measured assembly indices are required to determine the threshold for life

... Therefore, it conforms with the theoretical expectation and highlights a well-known mathematical property in data compression and complexity science: specifically, that different parsings (of an object) can perform equally in terms of compression rate. This directly reveals that the illustrative examples presented in later work 28 fail to address our results, and further attempts 29,30 seem to overlook the intrinsic deficiencies (both theoretical and empirical) in AT demonstrated in the present article. ...

Experimental Measurement of Assembly Indices are Required to Determine The Threshold for Life

... Defining and quantifying complexity is something of debate, and several molecular complexity metrics have been proposed (e.g., Hazen et al., 2007;Rouvray and Bonchev, 2003); however, none of these measurements are directly measurable experimentally. Assembly Theory (AT) is one such theory that investigates the smallest number of steps required to form a given product from a defined set of starting materials (Marshall et al., 2021;Marshall et al., 2017;Sharma et al., 2023), and is measurable spectroscopically (Jirasek et al., 2024). This theory is not specific and can be applied to a variety of object formation questions, among them the construction of molecules. ...

Investigating and Quantifying Molecular Complexity Using Assembly Theory and Spectroscopy
  • Citing Article
  • April 2024

ACS Central Science

... On the grounds that a physical fossil will always be more conclusive than a signal. The same objective underlies recent efforts to quantify time through spatial extensivity in materials as gauged by the Assembly Index [29]. The interaction between the interior mind and the exterior engineered artifact is ongoing: improvements in mind lead to modifications of design, whereas improvements in artifact lead to increases in the acquisition of skill and expertise. ...

Formalising the Pathways to Life Using Assembly Spaces

Entropy

... What has been reported are experiments confirming the existence of a threshold, where for molecular assembly (e.g. AI of covalently bonded molecules), the threshold of AI greater than approximately 15 was identified when no abiotic samples tested exhibited a measured value above this threshold, but biological samples did [2][3][4][5][6][7][8]. The hypothesis of AT is that a threshold should exist, but the value of AI greater than approximately 15 for covalently bonded molecules was determined by experiment for those molecules with high enough abundance to be detectable. ...

Exploring and mapping chemical space with molecular assembly trees

Science Advances

... Complex or even relatively simple mixtures undergoing chemical transformations tend to combinatorically explode 19,[48][49][50] . Chemical reactants that can combine in multiple ways can form many products. ...

Identifying molecules as biosignatures with assembly theory and mass spectrometry

... What that threshold value might be, and whether the same value applies to chemical systems as different as organic and inorganic molecules, are questions deserving of additional study. Cronin, Walker, and colleagues et al. [1][2][3][4][5][6][7][8] have championed Assembly Theory (AT), which posits that two attributes serve to distinguish complex products of living systems from non-living objects: (i) living systems produce objects requiring numerous assembly steps and (ii) the objects are present in numerous copies. According to this hypothesis, only a directed assembly by life can overcome the combinatorial improbability of such complex objects. ...

Quantifying the pathways to life using assembly spaces

... It is now generally accepted [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] that information in the universe evolves, decreasing the information entropy . Assuming that the total entropy of the universe S is constant and is the sum of the information entropy and the physical entropy ℎ , we obtain [1] + ℎ = 0. ...

Defining Pathway Assembly and Exploring its Applications

... Contrary to the previous definition of the labeling map ϕ [1], the relation (4) preserves the commutativity of the assembly step but defines the order of concatenation of the strings, as -in (4) is superfluous if the vertices defining the directed edges of Ω are strings, as any edge e = (C x , C z ) unambiguously resolves The definition 1 is consistent: all vertices are unique (in any standard graph all vertices should be unique), and all are strings. Since an assembly step always consists of joining two parts only [4], this can be thought of as the left and right fragments of the newly formed string [3], and those strings that can be the result of concatenation of two shorter strings are assembly step 2-in-regular vertices, while unit-length strings are inaccessible. Remarkably, the uniqueness of each vertex is the sufficient criterion to establish if an assembly step is allowed (cf. ...

A probabilistic framework for identifying biosignatures using Pathway Complexity

... In the context of exoplanet searches for life, achieving this on a planetary scale requires the probabilistic search for anomalies that themselves have in-built structure. Marshall et al. (2017) have recently developed one possible complexity measure, they call Pathway Complexity, which quantifies the complexity of a any given object as the shortest pathway for its assembly. The measure identifies the shortest pathway to assemble a given object by allowing the object to be dissected into a set of basic building units, and rebuilding the object using those units. ...

A Probabilistic Framework for Quantifying Biological Complexity