Synthesis for Vesicle Traffic Systems

Abstract

Vesicle Traffic Systems (VTSs) are the material transport mechanisms among the compartments inside the biological cells. The compartments are viewed as nodes that are labeled with the containing chemicals and the transport channels are similarly viewed as labeled edges between the nodes. Understanding VTSs is an ongoing area of research and for many cells they are partially known. For example, there may be undiscovered edges, nodes, or their labels in a VTS of a cell. It has been speculated that there are properties that the VTSs must satisfy. For example, stability, i.e., every chemical that is leaving a compartment comes back. Many synthesis questions may arise in this scenario, where we want to complete a partially known VTS under a given property. In the paper, we present novel encodings of the above questions into the QBF (quantified Boolean formula) satisfiability problems. We have implemented the encodings in a highly configurable tool and applied to a couple of found-in-nature VTSs and several synthetic graphs. Our results demonstrate that our method can scale up to the graphs of interest.

Appendices

Appendix

A Discussion on the Choice of VTS Model

The molecules transported by the VTS are themselves its regulators. The molecules in a compartment/vesicle may be active or inactive. The molecules that are responsible for vesicle fusion are called SNARE proteins [8, 22]. Active SNAREs present on vesicles (v-SNAREs) bind with their cognate active SNAREs on the target compartment (t-SNAREs) to enable vesicle fusion. A cell contains multiple kinds of v- and t-SNAREs. Only specific pairs of v and t SNAREs can bind to each other and participate in fusion. Fusion compatible v- and t- SNAREs are determined by biological experiments. Different vesicle-compartment fusions in the cell are brought about by different v- and t-SNARE pairs. A molecule that participates in a given fusion reaction must not interfere with fusion at different compartments or vesicles. Therefore, SNAREs must be kept in an inactive form in appropriate compartments/vesicles. The activity of molecules is regulated by the other molecules, i.e., the presence and absence of the other molecules in a compartment or vesicle may make the molecule active or inactive. We call this regulation as activity functions. In the VTS model, we assume that the system is in steady state and the concentrations of the molecules in the compartments do not change over time. We define SNARE pairing specificity by a fusion pairing relation containing pairs of SNAREs and molecular regulation by activity Boolean functions. Since the system is in steady state, we expect that any molecule that leaves a compartment must come back via some path on the graph. We call this property of VTS as stability.

Our model is inspired by [9]. On the timescales of minutes, our following assumptions reasonably capture the important aspects of the Rothman-Schekman-Sudhof (RSS) model [23] of vesicle traffic system.

• 1. A cell is a set of compartments exchanging vesicles.

• 2. Compartments are neither created nor destroyed.

• 3. Each compartment is in steady state, gain and loss balance.

• 4. Molecules are neither created nor destroyed.

• 5. Molecules move via vesicles of uniform size.

• 6. Identical vesicles have identical target compartments.

• 7. Fusion of vesicles to compartments is driven by specific SNARE pairing.

• 8. The activity of a SNARE can be regulated by other molecules present on the same compartment or vesicle.

• 9. An active SNARE pair is necessary and sufficient for fusion.

SNARE proteins are the agents of vesicle fusion in eukaryotic cells. When SNAREs on vesicles (v-SNAREs) encounter their cognate SNAREs on target compartments (t-SNAREs), they form SNARE complexes [8], and a single SNARE complex releases enough energy to enable membrane fusion [24]. SNAREs are identified by the presence of a conserved 60–70 stretch of amino acids called the SNARE motif. Based on amino acid sequence, SNARE motifs fall into 4 classes: Qa, Qb, Qc, and R [8]. Across all intracellular vesicle fusion reactions, the associated SNARE complexes contain one of each of the four kinds of SNARE motifs; the v-SNARE contributes a single SNARE motif, usually it is an R-SNARE (although, exceptions are known: Sec22b and Ykt6 are both R-SNAREs which form parts of t-SNAREs [25]) and the rest of the three SNARE motifs are contributed by the t-SNARE. In the cell, different vesicle fusion reactions are associated with distinct v- and t-SNARE pairs.

The paper [9] consider three Q SNARES as a single molecule, we have extended this model by considering each complex molecule as distinct. In contrast to the [9], we allow Q and R-SNARE type distribution across the whole system to be uneven. In our model fusion is driven by an active combination of three Q SNARE and one R SNARE molecule. We have relaxed the pairing matrix constraint to comply with this fact. For biological efficiency and optimality reasons, we do not allow self-edges to be present in the VTS.

Fig. 1.

A found-in-nature VTS. Nodes and edges are labelled with sets of molecules. $\hat{}$ indicates that the molecule is active.

B The Natural VTSs

Here we will present the two VTS collected from the literature.

1.1 B.1 Mammalian VTS

The Fig. 1 represent mammalian SNARE map created by studying the wide array of literature. To construct the map, we have assumed that vesicles only contain a single active v-SNARE, and we have attributed t-SNAREs and inactive v-SNAREs that travel between compartments to one of the known vesicles that go between the same source and target compartments. In order to identify the active SNARE complex involved in any particular vesicle fusion, we used two criteria. The SNARE complex is formed in vivo. In most papers, this is determined by immunoprecipitation of the SNARE complex from the relevant cell fraction. Blocking SNARE complex formation (for example, using antibodies against these SNAREs, or using cytosolic forms of these SNAREs) blocks the specific transport step. Note that these vesicles have been collected from multiple cell types, and any given cell type is likely to contain only a subset of the vesicles in the map.

In this figure, the rectangles represent compartments, the identities of compartments are written within ER = endoplasmic reticulum, ERGIC = ER-Golgi intermediate compartment, RE = recycling endosome, EE = early endosome, LE=late endosome, LYS = lysosome, PM = plasma membrane. The arrows represent vesicle edges.

Fig. 2.

Yeast VTS

Fig. 3.

Run-times for synthesis queries. #C stands for minimum changes in the synthesized VTS in comparison with the given partial VTS. Time is reported in seconds. (a) The solver used is DepQBF (b) The solver used is Z3. The sub-mammal is a subgraph of the complete mammal-graph. In the Add/Delete parts column, ‘+’n sign is used to show the addition of n number of the molecules, similarly ‘-’n is used to show the removal of n number of molecules. In the table, N is node labels, AN is active node molecules, E is edges, PE is molecule presence on the edge and AE is active molecules on the edge. The [kC] stands for k graph connectedness which is part of only DepQBF experiments.

1.2 B.2 Yeast VTS

In Fig. 2, we present the yeast VTS. We have borrowed the VTS from [14]. It has been adapted from the paper by separating the v and the t SNAREs. It is clear that it is an incomplete description of the VTS. For example, the inactive molecules were not reported in the reference. We are currently searching for more literature that can help us complete all known information about the VTS.