Modeling the Geometry of the Endoplasmic Reticulum Network

Abstract

Abstract. We have studied the network geometry of the endoplasmic reticulum by means of graph theoretical and integer programming models. The purpose is to represent this structure as close as possible by a class of finite, undirected and connected graphs the nodes of which have to be either of degree three or at most of degree three. We determine plane graphs of minimal total edge length satisfying degree and angle constraints, and we show that the optimal graphs are close to the ER network geometry. Basically, two procedures are formulated to solve the optimization problem: a binary linear program, that iteratively constructs an optimal solution, and a linear program, that iteratively exploits additional cutting planes from different families to accelerate the solution process. All formulations have been implemented and tested on a series of real-life and randomly generated cases. The cutting plane approach turns out to be particularly efficient for the real-life testcases, since it outperforms the pure integer programming approach by a factor of at least 10.Keywords: endoplasmic reticulum, plane graph, 0-1 programming, separation procedure ProblemThe endoplasmic reticulum (ER) is a membrane-bound organelle that forms a highly complicated interconnected network of tubules and flattened sacs (known as cisternae) [10,6]. As the cortical ER in plant cells occupies a very thin, almost two-dimensional, layer of cytoplasm beneath the plasma membrane, our study of the ER network will be based on 2D approximations of the ER network in Tobacco leaf epidermal cells. Figure 1 (a) shows an instance of live cell images of an ER network [13]. Transition between tubules and cisternae can be highly dynamic and tubules can also dynamically change their polygonal network [10]. The dynamic ER shape is suggested to be adaptable to the cells requirements for ER function; for example, ER cisternae may be the preferred site of protein translocation while tubules might be the preferred site for ER vesicle budding. As an expanding number of proteins have been identified that mediate the generation and shape of the ER network, the stage is now set for investigations into the mechanisms regulating ER morphology within the cell. To be able to carry out such investigations, better tools are required to quantify the morphology and dynamic rearrangements that the ER undergoes. It is important to consider that whilst the proteins and stresses on the system driving formation and changes in the remodelling may differ between eukaryotic systems, the network geometry of the ER appears to be fairly well conserved in terms of it consisting of a polygonal network of interconnected tubules and cisternae. Therefore, the problem considered here has universal appeal towards understanding the constraints placed on ER network formation in all systems.A quantitative analysis [11] of the ER network in tobacco leaf epidermal cells suggests that it is a perturbed Euclidean Steiner network between terminals, where terminals are persistent nodes (static elemen...

Modeling the Geometry and Dynamics of the Endoplasmic Reticulum Network

“…θ = 180 o ) and the number of closest neighbours to take into account (20 in the paper). This choice of 20 neighbours is sufficient to produce the optimal solution satisfying the angle constraint for the real-life test cases [14]. Comparatively, we extend the optimization problem in the following aspects:…”

“…Thus, a degree constraint for end nodes might be necessary. In the following we illustrate a modified optimization problem from [14] for constructing the network topology and the dynamics for each type of nodes.…”

“…TABLE 1 Average number of angle constraints according to the number of neighbours considered and the angle threshold θ for a set of 50 real-life test cases. Column "BN only" corresponds to the number of constraints added in the full model of paper [14] when angle constraints are applied to branching nodes only, with a neighbourhood threshold of 20. The last column shows the number of equivalent constraints generated with the lifting technique of section 3.1…”

“…In [14], we have formulated an optimization problem to construct the network topology with degree constraints (either degree 3 or degree no more than 3) and angle constraints for given degree-3 nodes. Note that in the ER network dynamics, the set of degree-3 nodes may change as they may include some persistent nodes.…”

“…Note, that the local optima might not give a unique network topology. In [14] we have proposed an optimization problem to construct an optimal network minimizing the total length with constraints on angles and degrees of nodes; such optimal networks are globally optimal and are close to the ER networks from experimental data.…”