Golgi compartments enable controlled biomolecular assembly using promiscuous enzymes.

The synthesis of eukaryotic glycans - branched sugar oligomers attached to cell-surface proteins and lipids - is organized like a factory assembly line. Specific enzymes within successive compartments of the Golgi apparatus determine where new monomer building blocks are linked to the growing oligomer. These enzymes act promiscuously and stochastically, causing microheterogeneity (molecule-to-molecule variability) in the final oligomer products. However, this variability is tightly controlled: a given eukaryotic protein type is typically associated with a narrow, specific glycan oligomer profile. Here, we use ideas from the mathematical theory of self-assembly to enumerate the enzymatic causes of oligomer variability and show how to eliminate each cause. We rigorously demonstrate that cells can specifically synthesize a larger repertoire of glycan oligomers by partitioning promiscuous enzymes across multiple Golgi compartments. This places limits on biomolecular assembly: glycan microheterogeneity becomes unavoidable when the number of compartments is limited, or enzymes are excessively promiscuous.

Introduction

The surfaces of all living cells are decorated with information-rich oligosaccharide molecules known as glycans: branched sugar oligomers covalently linked to proteins or lipids (Varki et al., 2017). Glycans encode cell identity and mediate a variety of intercellular interactions. They play critical roles in development, species recognition, self-nonself discrimination, and host-pathogen coevolution (Varki, 2017).

Glycans are composed of a small set of monosaccharide building block types (monomers) and disaccharide bond types (linkages in the branched glycan oligomer) (Adibekian et al., 2011; Figure 1A). Eukaryotic glycans are built by collections of glycosyltransferase (GTase) enzymes the ER and Golgi apparatus, a process known as glycosylation. GTase enzymes are chemically precise but contextually sloppy (Taniguchi and Honke, 2014; Biswas and Thattai, 2020): a given enzyme catalyzes a specific bond between a specific pair of monomer types, but can act promiscuously and stochastically on many oligomer types (Figure 1B). As a consequence, even a single cell with a limited set of GTase enzymes can theoretically synthesize an astronomical array of oligomeric combinations (Cummings, 2009).

Figure 1.

Glycan synthesis by promiscuous enzymes in Golgi compartments.

(A) A GTase enzyme catalyzes a glycosidic linkage between a specific donor monomer type (the ‘donor-substrate’) and a specific acceptor monomer type with specific branches or roots (the ‘acceptor-substrate’). We represent distinct monomer types in an oligomer by shapes/colors, and linkages between distinct monomer carbons by distinct bond angles (Varki et al., 2017). (B) GTase enzymes can show varying degrees of promiscuity, demonstrated here by three real examples of enzymes that act on galactose acceptors (Taniguchi and Honke, 2014, Chapter 39,44,47). We represent each GTase enzyme graphically, showing its acceptor-substrate and the specific monomer-addition reaction it catalyzes (red arrow from acceptor to donor at distinct angles for distinct acceptor carbons). The root-sensitive enzyme Fut2 requires the galactose acceptor to be linked to a GlcNAc root monomer. The branch-sensitive enzyme GTB requires its galactose acceptor to have a fucose branch. The branch-insensitive enzyme β4GalNT1 can act on branched or un-branched galactose monomers. (C) The localization of enzymes in successive Golgi compartments can sometimes mitigate the variability caused by enzyme promiscuity. This is demonstrated using the example of blood group B-antigen synthesis by the root-sensitive (RS) enzyme Fut2 and branch-sensitive (BS) enzyme GTB. We consider hypothetical scenarios in which these enzymes are context-free (CF). Increased enzyme promiscuity leads to tandem repeat byproducts. Placing the context-free Fut2 enzyme in a separate compartment from branch-sensitive GTB restores specific synthesis of the B-antigen. (D) Our main result is illustrated by this example (elaborated in Figure 4). We show the minimum number of compartments needed to specifically synthesize the same product oligomer from any of its sub-oligomers. We compare three broad classes of enzymes: ideal root-sensitive and branch-sensitive enzymes (ideal enzymes can read root chains or branches to arbitrary depth) or context-free enzymes. Ideal root-sensitive enzymes can distinguish the position of every acceptor monomer in an oligomer, so they can specifically synthesize any product in a single compartment. Branch-sensitive and context-free enzymes are more promiscuous; splitting such enzymes across multiple compartments increases the repertoire of oligomers they can specifically synthesize.

Figure 4.

The glycan biosynthetic repertoire of multi-compartment systems.

(A) Outline of Theorem 1 (Appendix 2). Suppose we are given a series of compartments that specifically synthesizes a desired target oligomer from a given input oligomer. We now proceed to modify the original compartments through several steps. Step I → II: Ensure enzyme concentrations are high, to eliminate intermediate oligomers. Step II → III: Remove all enzymes with acceptor blocks, to eliminate divergent reactions. Step III → IV: Replace each compartment with a series of infinite-residence-time compartments each containing a single enzyme. At each step, at least one of the original growth orders still remains, leading to the desired target oligomer. (B) We are given a target oligomer (bottom). We list all its sub-oligomers, and connect two oligomers by a green edge if some single-enzyme compartment can specifically synthesize the lower oligomer from the higher oligomer. All single-monomer additions can be achieved with ideal root-sensitive enzymes; only a subset of these can be achieved with branch-sensitive or context-free enzymes. To find the minimum number of compartments required to synthesize the target oligomer from any sub-oligomer, we decompose allowed paths into algorithmic growth stretches (Figure 1D).

A given glycosylated protein type in a given cell is usually associated with multiple glycan oligomers, a form of molecule-to-molecule variability known as microheterogeneity. The enormous potential for variability is exemplified by prokaryotic glycans, which are typically random heteropolymers (Varki et al., 2017, Chapter 21,22; Adibekian et al., 2011). In contrast, eukaryotic glycoproteins are typically associated with a narrow, specific set of glycan oligomers, referred to as protein’s glycan profile (Varki et al., 2017, Chapter 1; Campbell et al., 2014). The low variability of eukaryotic glycans is functionally relevant: specific protein glycan profiles are associated with distinct species (Hamako et al., 1993; Hård et al., 1992; Hokke et al., 1994), distinct individuals (as with ABO blood groups [Patenaude et al., 2002]) and distinct cell types in an individual [West et al., 2010]; and altered glycan profiles are implicated a variety of disorders (Freeze and Ng, 2011; Gill et al., 2013; Lo-Guidice et al., 1994).

How do eukaryotic cells generate narrow, specific protein glycan profiles despite the variability caused by promiscuous enzymes? Simulations suggest that the localization of enzymes within Golgi compartments plays a key role (Liu et al., 2008; Spahn and Lewis, 2014; Spahn et al., 2016; Fisher et al., 2019). Consider the GTases Fut2 and GTB, which specifically synthesize the blood group B-antigen (Taniguchi and Honke, 2014, Chapter 44,47). Hypothetical promiscuous versions of these enzymes, if placed in a single compartment, generate variable tandem repeat oligomers as byproducts (Figure 1C); specific synthesis of the B-antigen is restored when the promiscuous enzymes are placed in two separate compartments. However, tandem repeat synthesis is just one among many sources of variability, and GTases can be more or less promiscuous than the enzymes in our example. We would like to understand the general conditions under which compartmentalization mitigates glycan variability.

Mechanisms for controlling variability are central to the field of algorithmic self-assembly, which explores how building blocks with sloppy interactions can be programmed to assemble into a desired final structure (Soloveichik and Winfree, 2007; Zeravcic and Brenner, 2014; Murugan et al., 2015). Here, we use ideas from self-assembly theory to precisely enumerate the possible sources of glycan variability. Using rigorous mathematical theorems, we demonstrate that compartmentalization mitigates variability for large classes of promiscuous enzymes (Figure 1D) (proofs of theorems are provided in Appendix 2). Since the number of distinct Golgi compartments in any cell is limited, microheterogeneity due to byproduct synthesis becomes unavoidable during the synthesis of complex oligomers. Nevertheless, multi-compartment synthesis greatly increases the set of oligomers eukaryotes can specifically synthesize (Figure 1D), complementing other mechanisms of enzymatic control such as kinetic regulation. Thus, a quintessential eukaryotic trait (intracellular compartments) provides the means to overcome a fundamental biochemical limitation (enzyme promiscuity). This capability may underlie the conservation of the Golgi across eukaryotes (Barlow et al., 2018): the idea of the Golgi apparatus as a factory assembly line is more than a metaphor, it is a mathematical and biological necessity.

Results

Glycan synthesis in Golgi compartments

We focus on the diverse class of O-glycans, which are associated with most eukaryotic cell-surface proteins (Varki et al., 2017, Chapter 10). The synthesis of O-glycan oligomers begins in the Golgi apparatus, when a root monomer is attached to a specific serine or threonine on a substrate protein. The Golgi consists of an ordered series of compositionally distinct compartments (for example, cis, medial and trans cisternae). Each compartment contains a specific set of GTase enzymes responsible for growing an oligomer on a specific protein type. Multiple models are proposed for how oligomers and enzymes are trafficked through the Golgi. In the transport model, oligomers are ferried through successive compartments; in the maturation model, oligomers remain in place while the enzymatic composition of the compartment undergoes a rapid switch-like transition between successive states (Mani and Thattai, 2016; Pantazopoulou and Glick, 2019). Under either scenario, whether transport or maturation, growing oligomers spend time within successive enzymatic compartment types. (The residence time of an oligomer within a compartment type is exponentially distributed in transport model, and sharply peaked in the maturation model; our results only depend on whether the average residence time is small or large compared to the average time for monomer addition.) The oligomer finally exits the last compartment in the series. The set of final oligomers associated with a given substrate protein makes up its glycan profile.

GTase promiscuity and stochasticity

As the growing oligomer spends time within successive Golgi compartments, it encounters distinct collections of GTase enzymes (Moremen et al., 2012; Pantazopoulou and Glick, 2019). During each such encounter, the enzyme scans the oligomer for a site that matches a structural motif (the ‘acceptor-substrate’) and attaches a single free monomer (the ‘donor-substrate’) to that site (Figure 1A; Liu et al., 2008; Moremen et al., 2012). A given GTase enzyme invariably catalyzes a glycosidic (C-O-C) linkage between specific carbons on a specific free donor monomer type and a specific acceptor monomer type within the acceptor-substrate (Figure 1A). However, these enzymes are doubly sloppy: they are both promiscuous and stochastic.

Promiscuity

There are two strong reasons to expect GTase enzyme promiscuity. First, there are far fewer GTase enzymes than observed glycan oligomers in any given species (Narimatsu et al., 2017); moreover, the same highly conserved enzymes are found across species with very distinct glycans (Kaneko et al., 2001; Figure 2). Therefore, by the pigeonhole principle, an individual GTase enzyme must be promiscuous and able to act on many distinct oligomer types (Narimatsu et al., 2017). Second, although oligomers can be arbitrarily large, there is a limit to the size of the motif any enzyme can recognize within it. A given enzyme could act everywhere its recognition motif is found on an oligomer; the smaller the recognition motif, the more oligomers it will be found within, and the more promiscuous the enzyme (Blixt et al., 2008; Taniguchi and Honke, 2014; Biswas and Thattai, 2020). In our analysis we consider three broad classes of enzymes, exemplifying varying degrees of promiscuity (Figure 1B). Context-free enzymes are those whose acceptor-substrates are single monomers, no matter what branches or roots they are linked to. These enzymes are maximally promiscuous. Ideal branch-sensitive enzymes are those whose acceptor-substrates are some acceptor monomer type having or lacking specific branches. These are intermediate in promiscuity. Ideal root-sensitive enzymes are those whose acceptor-substrates are some acceptor monomer type linked to a specific root chain. These enzymes can distinguish every monomer in an oligomer and are minimally promiscuous. Enzymes that read roots and branches to partial depth represent more complex types of intermediate promiscuity; we do not discuss these possibilities here.

Figure 2.

Observed patterns of glycan variability.

(A) We compare oligomer profiles from three datasets (Lo-Guidice et al., 1994; Hård et al., 1992; Hokke et al., 1994); observed oligomers are starred, only non-fucosylated oligomers are shown. For each profile, we show a hypothetical set of compartmentalized enzymes leading to its synthesis; cyclic arrows represent runaway reactions, blunt arrows represent the action of one enzyme blocking the action of another. Each enzyme set is associated with a reaction network; red arrows show single-monomer-addition reactions. All oligomers exit the compartment as outputs after some average residence time. The residence time of the last compartment in each series is ; the residence time of all other compartments is . (B) Effect of compartment residence time on glycan variability, quantified using Shannon entropy. The entropy of the compartment’s output distribution depends on that of its input distribution, on the structure of the compartment’s reaction network, and on its average residence time .

Stochasticity

Chemical reactions within cells are necessarily stochastic, due to the low molecule numbers involved. While the growing oligomer is within a given compartment, the order in which it encounters the available GTase enzymes is equivalent to a Markov process of random sampling with replacement, with randomly distributed time intervals between successive encounters (Gillespie, 1977). The reaction network of the compartment shows every possible oligomer growth order starting from a given input oligomer, as a result of all possible enzyme-catalyzed single-monomer-addition reactions in all possible permutations (Figure 2A). Since O-glycan oligomers are not pruned, these reactions are irreversible (Varki et al., 2017, Chapter 6). Within a reaction network, intermediate oligomers are those that can potentially be further extended by some available GTase enzyme, and terminal oligomers are those that cannot be further extended. Two identical input oligomers might take different paths in the reaction network as they encounter GTase enzymes in different random permutations and at different times. An oligomer might encounter the same enzyme repeatedly (if the enzyme is at high concentrations), or it might exit the reaction compartment without ever encountering some enzyme (if the enzyme is at low concentrations).

Patterns of glycan variability

Consider any assembly-line reaction in which oligomers are built by adding one monomer at a time. We can show that there are precisely three ways for variability to occur (Appendix 2, Remark 1): truncated, runaway, and divergent reactions. Each type of reaction variability corresponds to a specific output pattern that has been observed in real glycan datasets (Figure 2; Table 1; Varki et al., 2017, Chapter 1). These are exemplified by oligomer profiles of respiratory mucins from a cystic fibrosis patient (Lo-Guidice et al., 1994), human chorionic gonadotropin (CG) from a cell line (Hård et al., 1992), and horse chorionic gonadotropin (Hokke et al., 1994) (datasets from UniCarbKB [Campbell et al., 2014]). These patterns can be explained by the following enzymatic causes. For the mucins, two identical input oligomers exit the reaction compartment at different stages of growth; since some reactions are truncated, this gives a combination of both intermediate and terminal oligomers as outputs. For horse CG, a compartment contains two enzymes that drive a runaway reaction; this gives oligomers with an arbitrary number of tandem repeats. For both mucins and human CG, two enzymes compete for the same acceptor-substrate; this sets up a divergent reaction, with mutually exclusive oligomer fates depending on the random order of enzyme action.

Table 1.

Variability caused by promiscuous, stochastic enzymes.

Observed product pattern	Type of reaction variability			Enzymatic cause
intermediate oligomers	⇔	truncated reaction	⇔	low concentration
tandem repeats	⇔	runaway reaction	⇔	linkage loop
mutually exclusive fates	⇔	divergent reaction	⇒	acceptor block

We model stochastic assembly-line reactions as continuous-time Markov processes with constant transition probabilities per unit time. This provides a probability distribution over each possible fate of the final output oligomer. The Shannon entropy of this output distribution in bits captures the variability of the glycan profile; approximately, it is the log-base-two of the number of distinct high-abundance oligomers. Each source of variability makes a distinct contribution to the residence-time-dependent entropy, as seen for the hypothetical reaction networks in Figure 2. For horse CG, the input entropy of the final compartment is zero (since there is a unique input oligomer); for human CG and mucins the input entropy is 1.5 (since input oligomers are in a 1:1:2 ratio due to divergent reactions in earlier compartments). At short residence times, the entropy initially rises due to the exit of intermediate oligomers from truncated reactions. This corresponds to the mucin profile. At long residence times multiple intermediate oligomers converge to a few terminal oligomers, so the entropy decreases; this corresponds to the human CG profile. For the horse CG profile, the entropy stays high even at long residence times, due to synthesis of tandem repeats via runaway reactions.

Enzymatic causes of glycan variability

The examples discussed above suggest that each type of glycan variability is connected to a distinct enzymatic cause. We now make this connection rigorous (Table 1).

A truncated reaction causes intermediate oligomers to be produced as outputs. This occurs whenever the average waiting time for monomer addition (a quantity inversely proportional to enzyme concentrations) is comparable to or greater than the compartment’s residence time (Appendix 2, Remark 1). The only way all input oligomers are guaranteed to reach a terminal state (assuming no proofreading) is if enzyme concentrations are sufficiently high, or equivalently, the residence time is sufficiently long (schematically, ).

A runaway reaction is an infinite path in the reaction network, giving oligomers with arbitrary numbers of tandem repeats as outputs. This implies at least one enzyme must act repeatedly along a single chain. Ideal root-sensitive enzymes don’t permit runaway reactions, since their acceptor monomers are at a fixed depth from the root. To diagnose runaway reactions for branch-sensitive and context-free enzymes, we examine the compartment’s linkage network, summarizing the allowed order of monomer linkages (Figure 3A; Appendix 2, Remark 2). A compartment contains a runaway reaction if and only if its linkage network contains one or more loops (Appendix 2, Lemma 1), since each loop corresponds to a tandem repeat. Branch-sensitive GTase enzymes can prevent linkage loops by using triggers (branched acceptor-substrates that cannot be synthesized within the compartment from a single monomer; Figure 3D).

Figure 3.

Enzymatic causes of glycan variability.

(A) Runaway reactions occur whenever certain steps of oligomer growth can be iterated to produce tandem repeats. Loops in the linkage network are necessary and sufficient for runaway reactions (Appendix 2, Lemma 1). An acyclic linkage network imposes a partial order on monomer types along any chain. represents an arbitrary oligomer. Boxes with black triangles represent the linkage network, showing all orders in which monomer types can be linked, with arrows from acceptor monomer types to donor monomer types (Appendix 2, Remark 2). Double arrows in linkage and reaction networks represent multiple reaction steps. (B) Divergent reactions occur whenever the reaction network has a fork that can never reconverge. This occurs when the action of one enzyme blocks the subsequent action of another, so the fate of the oligomer depends on the random order of enzyme action. Blunt red arrows in the linkage network represent the action of one enzyme blocking the action of another. Acceptor blocks are necessary (but not sufficient) for divergent reactions (Appendix 2, Lemma 2). Unidirectional acceptor block (only branch-sensitive enzymes): the acceptor-substrate of one enzyme is on some branch of the acceptor-substrate of another. Bidirectional acceptor block (all enzyme classes): two enzymes compete for the same acceptor-substrate. (C) We start with a compartment containing an arbitrary collection of enzymes. We can eliminate truncated reactions by ensuring high enzyme concentrations. We can eliminate runaway reactions by removing (or disabling with triggers, for branch-sensitive enzymes) at least one enzyme involved in each linkage loop. We can eliminate divergent reactions by removing all but one enzyme involved in each block. The result is an algorithmic compartment: for each possible input, it specifically synthesizes a corresponding unique output. (D) Properties of algorithmic growth, for different enzyme classes. See Appendix 3—figure 1 for a detailed example of branch-sensitive algorithmic growth.

Appendix 3—figure 1.

Efficient search for a branch-sensitive algorithmic growth order from input to target.

A divergent reaction is a fork in the reaction network that never reconverges, with distinct paths leading to mutually exclusive oligomer fates as outputs. To diagnose divergent reactions, we examine the acceptor-substrates of every enzyme in the compartment. A fork occurs whenever distinct enzymes can act on the same oligomer to yield distinct products. If these enzymes could act in any order (for example, if they act on distinct empty acceptor monomers on the oligomer) then the reaction paths could reconverge after the fork. If a fork does not reconverge this implies that the action of one enzyme on an oligomer blocks the subsequent action of another (Appendix 2, Lemma 2). For root-sensitive and context-free enzymes, this can only occur if both enzymes compete to act on the same carbon of the same monomer (bidirectional acceptor block; Figure 3B). For branch-sensitive enzymes, there are two ways this can occur: both enzymes have the same acceptor-substrate (bidirectional acceptor block; Figure 3B) or the acceptor-substrate of the first enzyme is on a branch of the acceptor-substrate of the second (unidirectional acceptor block; Figure 3B).

A compartment containing an arbitrary collection of enzymes might permit runaway and divergent reactions (Figure 3C, top). Our analysis shows that these sources of variability are not due to any individual enzyme: they arise from interactions within collections of enzymes (Figure 3A,B). We can eliminate truncated reactions by requiring high enzyme concentrations; we can eliminate linkage loops and acceptor blocks by removing certain enzymes from the reaction compartment (Figure 3C; Appendix 2, Lemma 3). These steps result in an ‘algorithmic compartment’: for each possible input, it specifically synthesizes a corresponding unique output. If we could watch an individual oligomer growing within an algorithmic compartment, we would find that its growth order always satisfied certain special properties that are easy to check, which we collectively characterize as ‘algorithmic growth’ (Appendix 2, Lemmas 4,5). The characteristics of algorithmic growth depend on the degree of enzyme promiscuity (Figure 3D). We now discuss how the concept of algorithmic growth can be used to find the minimum number of compartments required to specifically synthesize any desired oligomer.

Controlled glycan synthesis in multiple compartments

Suppose we want to specifically synthesize a given target oligomer from a given input oligomer, with no byproducts. As a first attempt, we might pick an arbitrary growth order that leads, one monomer at a time, from input to target. Each monomer-addition reaction corresponds to the action of some enzyme. We could simply load a single compartment with this set of enzymes. This guarantees that the target oligomer will be synthesized from the given input. The problem is, various byproducts might also be synthesized due truncated, runaway, or divergent reactions; and the target oligomer might itself be further extended at long residence times. Since these problems arise due to interactions within collections of enzymes, they might be avoided by splitting the enzymes across several compartments. The following theorems provide the answer to two questions (Figure 4A; Appendix 2): Is specific synthesis of the target from the given input even possible? If so, what is the minimum number of compartments required?

Theorem 1

A target oligomer can be specifically synthesized from an input oligomer if and only if it can be specifically synthesized from that input oligomer in a series of single-enzyme compartments.

Theorem 2

A target oligomer can be specifically synthesized from an input oligomer in a series of compartments if and only if there is a growth order from the input to the target that can be fully decomposed into algorithmic growth stretches.

Theorem 1 provides an efficient protocol to search for a solution: we needn’t consider all possible multi-enzyme combinations, it is sufficient to check single-enzyme compartments (Figure 4B). Once a solution is found using single-enzyme compartments, Theorem 2 provides a protocol to construct a solution using fewer multi-enzyme compartments. In this way, we are guaranteed to find the minimum number of compartments required for specific synthesis of the target (assuming a solution exists). For ideal root-sensitive enzymes a single compartment is always sufficient, since every growth order to a given terminal oligomer is algorithmic (Appendix 2, Lemma 4). The minimum-compartment path could be longer for the more promiscuous branch-sensitive or context-free enzymes (Figures 1D and 4B). There are some situations in which glycan variability is unavoidable: certain oligomers can never be synthesized without byproducts, no matter how many compartments we allow. In Appendix 3—figure 1 we discuss an example of a target oligomer that cannot be synthesized using branch-sensitive enzymes in fewer than two compartments.

Discussion

Living systems excel at building complex structures using stochastic, unreliable molecular components. Macromolecules such as DNA, RNA and proteins are built by copying known target templates, so errors can be removed by proofreading (Murugan et al., 2012). However, most biological structures are encoded using a step-by-step recipe – that is, an algorithm – rather than a template (Navlakha and Bar-Joseph, 2011). This is apparent during animal development: the genome encodes a recipe to make an adult, it is not a homunculus of the adult. Eukaryotic glycan synthesis is an exquisite demonstration of the same concept: re-configurable reaction networks act as template-free recipes to specifically encode diverse glycan oligomers.

In the absence of a template, cells use a spectrum of mechanisms to limit glycan variability. At one end are quantitative kinetic mechanisms (Cardelli et al., 2018), such as the regulation of compartment residence times (Pantazopoulou and Glick, 2019), control of donor-substrate levels (Parker and Newstead, 2019), or use of heteromeric enzymes to enhance sequential reactions (Varki et al., 2017, Chapter 4). At the other end are qualitative mechanisms that constrain which biosynthetic reactions are even allowed to occur: the selectivity of enzymes for their oligomer substrates, and the compartmentalization of enzymes within the Golgi. When enzymes are promiscuous, the kinetics of distinct reactions cannot be independently regulated, leading to variability (Biswas and Thattai, 2020). We have shown that, by splitting promiscuous enzymes across different Golgi compartments, this variability can be reduced or eliminated.

To synthesize oligomers without byproducts, cells must either decrease enzyme promiscuity, or increase the number of compartment types (Figure 1D). Both these strategies come at a cost, because there is a limit to the number of proteins a cell can encode. If all enzymes were highly selective, each with one specific substrate, a single compartment would suffice; but cells would require as many enzymes as oligomer types. On the other hand, an extensive system of membrane traffic proteins is required to maintain distinct Golgi compartments (Pantazopoulou and Glick, 2019). How do cells manage this trade-off? Some GTase enzymes, such as those which synthesize the N-glycan oligomannose precursor in the ER, are highly selective and appear to act on just one or two oligomers; but most GTases are promiscuous, able to act on many oligomers or at many points on a single oligomer (Biswas and Thattai, 2020). This means having multiple compartments is crucial for limiting glycan variability in real cells. These ideas inform strategies for artificial glycan synthesis (Liu et al., 2019) and algorithmic self-assembly (Soloveichik and Winfree, 2007; Zeravcic and Brenner, 2014; Murugan et al., 2015): ‘stop-and-go’ (Liu et al., 2019) or ‘step-assembly’ (Doty, 2012) approaches, which are analogous to multi-compartment synthesis, expand the repertoire of synthesizable oligomers.

The regulation of glycan synthesis by compartmentalization has biological advantages. By redistributing enzymes within the Golgi, multicellular organisms can use the same set of enzymes to generate distinct glycan profiles in distinct cell types (Figure 2A; West et al., 2010; Fisher et al., 2019). The rapid and reversible changes in glycan profiles seen during infection and inflammation (Varki et al., 2017, Chapter 46) are more consistent with changes in enzyme localization than changes in enzyme expression. Such changes could also facilitate rapid evolution of glycan profiles in the context of host-pathogen interactions and speciation (Watanabe et al., 2020; Varki et al., 2017, Chapter 20). But this also means small errors in Golgi localization could be pathological: glycan perturbations promoting tumor invasiveness appear to arise due to errors in enzyme localization, not mutations in the enzymes themselves (Gill et al., 2013; and congenital glycosylation disorders are often correlated with defects in Golgi structure (Freeze and Ng, 2011).

The emergence of intracellular compartments was a watershed step in eukaryotic evolution (Dacks and Field, 2018). Many hypotheses have been advanced about the adaptive function of such compartments. Here, we have shown that the compartmental organization of the Golgi apparatus allows cells to control glycan synthesis despite enzymatic promiscuity, potentially explaining why this remarkable organelle is universally conserved across all extant eukaryotic lineages (Barlow et al., 2018). The ability to generate narrow, specific glycan profiles would have been advantageous to early eukaryotes (Wetzel et al., 2018), enabling the sophisticated intercellular interactions that underlie sex, cooperation and multicellularity.

Materials and methods

Methods are provided in the Appendices. Appendix 1: Definitions. Appendix 2: Proofs. Appendix 3: Detailed example of algorithmic growth.

In the interests of transparency, eLife publishes the most substantive revision requests and the accompanying author responses.

Acceptance summary:

The manuscript addresses the interesting question of how, despite the sloppiness and promiscuity of the glycoenzymes, a limited variety of glycans are produced in the Golgi. The authors develop a complete formal theoretical framework and show that the sub-Golgi compartmentalization is key. It allows distributing the enzymatic activities through multiple processive Golgi compartments which effectively reduces the variability in the glycan output.

Decision letter after peer review:

Thank you for submitting your article "Golgi compartments enable controlled biomolecular assembly using sloppy enzymes" for consideration by eLife. Your article has been reviewed by three peer reviewers, and the evaluation has been overseen by a Reviewing Editor and Naama Barkai as the Senior Editor.

The reviewers have discussed the reviews with one another and the Reviewing Editor has drafted this decision to help you prepare a revised submission.

Summary:

The manuscript addresses the question of the control of glycan variability in the Golgi achieved by sloppy glyco-enzymes from a theoretical perspective. The authors propose that the variability issue can be solved by a spatial segregation of the enzymes, which could a posteriori provide an explanation for the functional origin of the Golgi compartmentalization.

Essential revisions:

The three reviewers and myself think that your work could be of strong interest for biologists working on membrane traffic and specially on Golgi, but in its current form, the message and the predictions for practical tests on living cells are not easily accessible to the majority of eLife readers. Therefore, we consider that you should deeply revise your manuscript to address this serious concern and discuss a few practical cases. In addition, since the dynamical structure of the Golgi is still a matter of debate, we think that you should add a discussion on how your model depends on Golgi maturation, on the consequence of a continuous gradient vs discrete distribution of the enzymes, on the impact in a multiple-compartments vs. a single compartment system. Eventually, we would like that you discuss the robustness of the model to the "uniform depth-first growth" rule.

[Editors' note: further revisions were suggested prior to acceptance, as described below.]

Thank you for submitting your article "Golgi compartments enable controlled biomolecular assembly using promiscuous enzymes" for consideration by eLife. Your article has been reviewed by three peer reviewers, and the evaluation has been overseen by a Reviewing Editor and Naama Barkai as the Senior Editor. The reviewers have opted to remain anonymous.

The reviewers have discussed the reviews with one another and the Reviewing Editor has drafted this decision to help you prepare a revised submission.

Summary:

This manuscript addresses theoretically the question of the control of glycan variability in the Golgi achieved by sloppy glyco-enzymes. The authors propose that the variability issue can be solved by a spatial segregation of the enzymes, which could a posteriori provide some functional origin to the Golgi compartmentalization.

Essential revisions:

The reviewers have appreciated your effort to facilitate the accessibility of your work to biologists reading eLife articles in your revised version. Nevertheless, we think that your paper could have a much stronger impact among biologists if you could further modify the presentation. We recommend following suggestions from reviewer #2 (see below).

Reviewer #2:

What I like about the work is the element of bringing in a new dimension into the intricate biological processes in the Golgi that involve glycan synthesis, binding, and self-assembly. However, I find the choice of language in the manuscript seriously concerning. The authors have chosen to present their main idea as mathematical theorems supported by proofs and lemmata. If the aim for this choice were to give the readers the assurance that what they are seeing is unquestionable fact and not just the opinion or suggestion of theoreticians, I believe this could be done in a much more accessible way. If the aim were to purport that the authors are making a new contribution to the field of mathematical logic by proving previously unknown theorems, then the aim would be incorrect and inappropriate. This has already been done in the specific context of algorithmic self-assembly in the papers that the authors cite themselves. The authors are merely applying these ideas to the specific biological problem at hand.

The main tenet of the authors' argument is enumerating the number of mutually exclusive and complementary possibilities in polymerization. This is basically equivalent to saying that a coin has two sides. One can, of course, make this into a theorem in mathematical logic and provide a proof of it, which can take pages and pages. However, what a normal user of this theorem would want to know from it will only be that we have only two mutually exclusive possibilities when we are throwing a coin: heads and tails. If we throw a dice, we have six possibilities. That's it. I do not see why the authors have chosen the rather heavy mathematical language to motivate the use of this simple enumeration of possibilities, for this particular biological application.

In response to my previous recommendation to streamline the presentation and make it more accessible to the appropriate biological readership of the journal, the authors moved some of the theorems and proofs to the appendix. As a result, the main manuscript now starts from lemma 3 and then wanders into some other mathematical blocks along the way. The only biological analysis is done in the captions of the figures, which are excessively long. Figure 1's caption is 29 lines if I managed to count correctly.

I think as it stands the manuscript will likely not attract its full potential in readership because of this confused style. My suggestion to the authors is to remove the mathematical language altogether and write the manuscript as if they are applying the enumeration argument to the biological process. They can motivate it, provide the examples next, and conclude by arguing for the generality of the principles and the arguments. I imagine this would make a wonderful paper that is also going to be read by the readers of eLife.

I cannot recommend publication of the manuscript in its current style, not even with the revision.

Essential revisions:

The three reviewers and myself think that your work could be of strong interest for biologists working on membrane traffic and specially on Golgi, but in its current form, the message and the predictions for practical tests on living cells are not easily accessible to the majority of eLife readers. Therefore, we consider that you should deeply revise your manuscript to address this serious concern and discuss a few practical cases.

We have taken very seriously the input that our work should be more accessible to biologists. Before preparing our revision, we circulated the manuscript to cell biology colleagues to solicit specific feedback on (a) relevance of results and (b) clarity of exposition. Based on this feedback and on the reviewer comments, we have made major revisions to the paper, summarized here:

New emphasis: We received feedback that our analysis of promiscuity is potentially the most useful direct result of our work. We have therefore revised the manuscript to focus deeply on the causes and consequences of enzyme promiscuity. Both in the title and throughout the text, we have replaced the word “sloppy” (which was borrowed from the self-assembly literature) with the word “promiscuity” which is more informative and relevant to biologists. We have added several paragraphs to the Discussion, in which we outline the trade-off between promiscuity and compartmentalization.

New results: We have added completely new results, which compare the behavior of two classes of enzymes that span a range of possible promiscuity: the maximally promiscuous “branch-insensitive enzymes” and the less promiscuous “branch-sensitive enzymes”. This includes two new Lemmas (3 and 4) in Supporting Information. Our discussion is updated to show how promiscuity and compartmentalization jointly influence glycan variability, with several new predictions and new examples.

Modified figures: We have modified all figures (Figures 1, 2, 3 and Appendix 3—figure 1) to update our comparison of branch-insensitive and branch-sensitive enzymes. Our key new results are summarized in an example discussed in Figure 3C.

Streamlined text and terminology: We have replaced several technical terms with more informative terms and added multiple paragraphs that provide a non-technical background to the mathematical theorems. We have replaced the term “uniform depth-first growth” with “algorithmic growth” and expanded its definition to apply to both branch-insensitive and branch-sensitive enzymes (Lemma 4).

In addition, since the dynamical structure of the Golgi is still a matter of debate, we think that you should add a discussion on how your model depends on Golgi maturation, on the consequence of a continuous gradient vs discrete distribution of the enzymes, on the impact in a multiple-compartments vs. a single compartment system.

We thank the reviewers for pointing out this lacuna in our discussion. We have added new text (in the section “Glycan synthesis by promiscuous, stochastic enzymes”) that discusses in detail the implications of the transport and maturation models of the Golgi. We discuss why, as long as the maturation transitions are sufficiently rapid, our results will apply to both models. We have noted that our results will not apply to the case of a gradient of enzymatic compositions.

Eventually, we would like that you discuss the robustness of the model to the "uniform depth-first growth" rule.

We thank the reviewers for this insightful comment. Indeed, the definition of “uniform depth-first growth” is not universal but depends on the nature of the underlying enzyme promiscuity. In our revision we have explicitly derived independent results for two classes of enzymes that span a range of promiscuity: branch-insensitive and branch-sensitive enzymes. We have replaced the term “uniform depth-first growth” (which applies only to the latter class) with the broader term “algorithmic growth” and described the different types of algorithmic growth which apply to the different enzyme classes (Lemma 4). Our main result (Theorem 2) still applies, to both classes of enzyme promiscuity. Since real GTase enzymes will have intermediate promiscuity between the two classes, our results therefore provide both an upper and a lower bound on the number of compartments needed to specifically synthesize a given oligomer (Figure 3C). We therefore expect our predictions to be applicable in many cellular contexts.

[Editors' note: further revisions were suggested prior to acceptance, as described below.]

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Reviewer #2:

What I like about the work is the element of bringing in a new dimension into the intricate biological processes in the Golgi that involve glycan synthesis, binding, and self-assembly. However, I find the choice of language in the manuscript seriously concerning.

We thank the reviewer for this suggestion. We have extensively revised the figures and text of the manuscript. We have streamlined the flow of text to focus on just the key ideas and remove distracting technical asides.

The authors have chosen to present their main idea as mathematical theorems supported by proofs and lemmata. If the aim for this choice were to give the readers the assurance that what they are seeing is unquestionable fact and not just the opinion or suggestion of theoreticians, I believe this could be done in a much more accessible way.

This is an important point. We had chosen to present our results in the form of theorems, rather than the alternative possibility of simulations. Our aim was three-fold. First, we hoped to highlight general lessons, which is extremely difficult using simulations since any implementation of a simulation involves making several specific choices. Second, we wanted to ensure that all our assumptions were clearly stated, and that we did not miss any hidden assumptions about the behavior of enzymes. Third, we sought to provide a bridge to self-assembly researchers, who we hoped would be stimulated by ideas derived from this compelling biological example of controlled synthesis. We hope our updated text, involving more examples to buttress the mathematics, is less distracting and provides a more accessible exposition of the central ideas.

If the aim were to purport that the authors are making a new contribution to the field of mathematical logic by proving previously unknown theorems, then the aim would be incorrect and inappropriate. This has already been done in the specific context of algorithmic self-assembly in the papers that the authors cite themselves. The authors are merely applying these ideas to the specific biological problem at hand.

Although the self-assembly literature has a rich body of mathematical theorems, to our knowledge there is no specific application of these ideas to the construction of tree-like objects, which glycans are an example of. The basic idea of variability (runaway and divergent reactions) has been previously explored. However, the theorems we have proved go beyond application of existing theorems, and to our knowledge they are not to be found elsewhere the literature.

The main tenet of the authors' argument is enumerating the number of mutually exclusive and complementary possibilities in polymerization. This is basically equivalent to saying that a coin has two sides. One can, of course, make this into a theorem in mathematical logic and provide a proof of it, which can take pages and pages. However, what a normal user of this theorem would want to know from it will only be that we have only two mutually exclusive possibilities when we are throwing a coin: heads and tails. If we throw a dice, we have six possibilities. That's it. I do not see why the authors have chosen the rather heavy mathematical language to motivate the use of this simple enumeration of possibilities, for this particular biological application.

One of our major results is, indeed, the enumeration of mutually exclusive possibilities. The reviewer is correct that the application of this result does not require the detailed machinery of the theorem. However, one of our key contributions (summarized Table 1) is to show that the possibilities we have listed are indeed comprehensive. To prove we have not missed out other sources of variability, we required the full logical machinery. However, we agree that these details need not be in the main text. We have restricted them to the Appendices.

In response to my previous recommendation to streamline the presentation and make it more accessible to the appropriate biological readership of the journal, the authors moved some of the theorems and proofs to the appendix. As a result, the main manuscript now starts from lemma 3 and then wanders into some other mathematical blocks along the way. The only biological analysis is done in the captions of the figures, which are excessively long. Figure 1's caption is 29 lines if I managed to count correctly.

We have made major changes to the flow of the text, with the inclusion of new examples, new figure panels, and re-ordering of key results. We have removed the distracting technical excursions, instead we now motivate the result with several examples, before moving on to their mathematical basis. We agree the figure captions were too long in the previous version. We have re-organized the text, so the captions contain only details needed to interpret the figures, while the text contains fuller discussion and examples.

I think as it stands the manuscript will likely not attract its full potential in readership because of this confused style. My suggestion to the authors is to remove the mathematical language altogether and write the manuscript as if they are applying the enumeration argument to the biological process. They can motivate it, provide the examples next, and conclude by arguing for the generality of the principles and the arguments. I imagine this would make a wonderful paper that is also going to be read by the readers of eLife.

We thank the reviewer for taking this broad view of the manuscript. We have now done exactly this, started with examples, provided some intuition for them, and concluded with arguments for generality. All the material needed to see what’s “under the hood” is still contained in the Appendices for the technical reader, and we have directed readers to the relevant Lemma or Theorem at appropriate points in the text.