The authors have declared that no competing interests exist.
Conceived and designed the experiments: PG AM. Performed the experiments: PG AM. Analyzed the data: PG AM. Contributed reagents/materials/analysis tools: PG AM. Wrote the paper: PG AM.
Cells of almost all solid tissues are connected with gap junctions which permit the direct transfer of ions and small molecules, integral to regulating coordinated function in the tissue. The pancreatic islets of Langerhans are responsible for secreting the hormone insulin in response to glucose stimulation. Gap junctions are the only electrical contacts between the beta-cells in the tissue of these excitable islets. It is generally believed that they are responsible for synchrony of the membrane voltage oscillations among beta-cells, and thereby pulsatility of insulin secretion. Most attempts to understand connectivity in islets are often interpreted, bottom-up, in terms of measurements of gap junctional conductance. This does not, however, explain systematic changes, such as a diminished junctional conductance in type 2 diabetes. We attempt to address this deficit via the model presented here, which is a learning theory of gap junctional adaptation derived with analogy to neural systems. Here, gap junctions are modelled as bonds in a beta-cell network, that are altered according to homeostatic rules of plasticity. Our analysis reveals that it is nearly impossible to view gap junctions as homogeneous across a tissue. A modified view that accommodates heterogeneity of junction strengths in the islet can explain why, for example, a loss of gap junction conductance in diabetes is necessary for an increase in plasma insulin levels following hyperglycemia.
Gap junctions are clusters of intercellular channels between cells formed by the membrane proteins connexins (Cx), that mediate rapid intercellular communication via direct electric contact and diffusion of metabolites
Figure credit: Mariana Ruiz LadyofHats,
As with many other excitable cells, the information content of bioelectric signals
A paradigm that is gaining increasing recognition is that bioelectric and (epi-)genetic signaling are related as a cyclical dynamical system
Competition between prevalent strategies and adaptive changes at the individual level characterize the sociologically motivated model of
Gap junctions are known to adapt on at least two timescales: trans-junctional currents are gated on a fast timescale of the order of a few milliseconds to seconds in response to a trans-junctional voltage difference (
Steady-state junctional currents from HeLa-Cx36 cell pairs indicate conductance,
Interestingly, the voltage-gated gap junction appears to conform to a homeostatic principle with respect to transjunctional current,
Our starting point is a model of competitive learning introduced in
There are few general principles that can organize an argument to discuss plastic behavior in excitable cells; Hebb’s postulate is one such. In common colloquialism this learning rule is stated as “cells that fire together, wire together”; in other words, temporal association between pairs of firing neurons is successively encoded in synaptic coupling between those neurons. A Hebbian philosophy asserts that the direction of adaptation is such as to reinforce coordinated activity between cells. One can now set forth some rules governing the above weight changes, which may have a Hebbian or anti-Hebbian flavor as the situation demands, and depend on the outcomes of the surrounding
We now consider a mean-field version of the model. The idea behind the mean-field approximation is that we look at the average behavior in an infinite system. This, at one stroke, deals with two problems: first, there are no fluctuations associated with system size, and second, the approximation that we have made in ignoring the “self-coupling” of the gap junction is better realized. In the mean-field representation, every gap junction is assigned a probability (uniform over the lattice) to be either strong (
To design a transition rule for gap junctions that is consistent with a Hebbian theory, and at the same time tunes gap junctional plasticity to voltage activity in the network, we mimic the homeostatic adaptation implicit in (fast) voltage-gating of conductance (
We write equations for the probability
The first term on the right hand side represents the probability that the strong state at time
We now write down all possible scenarios for
For example: if
Strong | Strong | |
Strong (Weak) | Weak (Strong) | |
Weak | Weak |
This evolution equation thus embodies that if
The steady-state distribution of weak and strong junctions is obtained as the fixed point solution of Eq. (2):
The physically reasonable condition on the firing probabilities is
The physically relevant (
For low firing probabilities, such as for example
Beta-cells were initialized as firing (1) or not (0), and gap junctions as weak (0) or strong (1) with equal probability. 5000 beta-cell–gap junction pairs (Fig. 3) were iterated according to the learning rules described in the text. The legend indicates the (
We see thus that similar behaviour for the two gap junctions induces strengthening, while dissimilar behaviour induces weakening, in line with the Hebbian viewpoint adopted above.
One major interest in developing a theory of gap junction adaption is to understand the changes in junctional conductance that take place in type 2 diabetes. It has been suspected from animal studies that loss of Cx36 is phenotypically similar to a prediabetic condition characterized by glucose intolerance, diminished insulin oscillations and first and second phases of insulin secretion, and a loss of beta-cell mass
A word about dimensionalities – while we recognise that the geometries of real synaptic networks are complex and that they are embedded in three dimensions, our choice of working in one dimension is based as much on simplicity as on the absence of a reason to choose a more complex geometry. Working on a three-dimensional lattice would only increase the complexity of our algebra, while not really getting closer to the real geometry of synaptic networks, which are, as the name suggests, probably embedded on abstract graphs. However, the fact that we have worked in mean field (ignoring correlations and going to the limit of infinite systems) in a one-dimensional embedding makes our results less reliant on the embedding geometry than they otherwise might have been. We mean by this that while specific quantitative estimates might well be affected by the inclusion of more neighbours in higher dimensionalities, the qualitative outlines of our calculations will remain very similar. Our choice of mean field dynamics both in this case (as well as in the original learning model of
The game-theoretic formalism presented here provides a high-level explanation why a loss of junctional conductance would be necessary in diabetes. In the healthy individual insulin secretion occurs relatively sparingly, for a few hours at regularly spaced intervals following glucose ingestion (breakfast, lunch and dinner). The low firing rates in a healthy individual are accompanied by a high proportion of strong gap junctions (that is, near the region marked by A,
In this way, the islet is able to accommodate a stimulus stronger than that for which its physiology had evolved. A change in
At the heart of our game-theoretic theory is its use of stochasticity in gap junction synchronisation. Classically, strong gap junctions entrain beta-cells to fire, the entire assembly is assumed to be fairly homogeneous in gap junction strength, and the resultant synchronous bursting is seen to be essential to GSIS. Our theory on the other hand, introduces the possibility that beta-cells coupled even to strong gap junctions may not fire, and likewise, weak gap junctions may induce simultaneous firing. Further, synchronous bursting, as well as the simultaneous
In principle it is possible to explain observations of junctional strengths such as in
We have constructed a theory that offers an alternative explanation to the classical view that gap junctions primarily function to synchronize beta-cells in an islet so the entire islet behaves like a syncytium and a uniform period emerges. When gap junction adaptation is considered, partial synchronization can occur even in networks fully coupled with (strong) gap junctions. This learning framework predicts in a natural fashion that a full synchrony across the islet is very unlikely, that synchronization is a local phenomenon and happens across a few groups of cells. Thus the view that emerges instead is that the islet is sensitive to a glucose demand in secreting insulin and uses gap junctions as a tuning parameter in this adaptation. Paradoxically, an increase in secretion efficiency can come not by strengthening junctions, but down-regulating them instead. Thus, a lowered conductance need not necessarily be interpreted as “failing” gap junctions. On the contrary, they are judiciously adapting to the increased glucose load to cope with an increased demand for insulin secretion.
At the moment there does not seem to be direct experimental evidence that a reduction of gap junctions occurs in human type 2 diabetes. Additionally, although it is very attractive from a theoretical viewpoint, it is not proven that gap junctions are altered in response to altered islet firing activity in diabetes. Our model is a complementary line of evidence, albeit theoretical, in these directions. Further, the model makes another related prediction, that gap junction expression and coupling strength are very likely to occur as heterogeneous across the islet, in both health as well as diabetes. If the naturally heterogeneous nature of gap junctions is acknowledged, this could be critical in designing appropriate clinical interventions, since connexins are potential targets for diabetes therapy. Indeed, we hope that our work will be helpful to researchers seeking to clarify the adaptive dynamics of gap junctions in diabetes.