Moving through the matrix
March 1, 2011 § Leave a comment
Blood vessel formation is one of the wonderful adaptive processes in biology. If a tissue is under-oxygenated, it sends out a cry for help and lo and behold, a new blood vessel forms. This is great if the rescued tissue was under-oxygenated because it was cut off from its normal supply by a wound. It’s not so good if the under-oxygenated tissue is a tumor. Tumors that successfully acquire a blood supply of their own can metastasize to different sites by travelling through the circulatory system, and grow much faster than avascular tumors.
So how do the new blood vessels actually form? In the context of a tumor, what happens is roughly this: the tumor sends out protein signals such as VEGF (vascular endothelial growth factor), which diffuses through the tissue until it reaches an existing blood vessel. The endothelial cells that line a blood vessel have receptors for VEGF, and they react to it by producing proteases that chew up the basement membrane that supports the blood vessel. The freed endothelial cells are then able to migrate into the extracellular matrix, again often chewing their way along using proteases such as matrix metalloproteinases. VEGF induces both chemotaxis and proliferation, so the new “sprout” of the blood vessel moves towards the tumor cell (heading up the gradient of VEGF) creating a column of endothelial cells that will later become hollow, grow basement membrane around it, and become able to support blood flow to the tumor. Presto, new blood vessel. In fact, many new blood vessels: the original sprout will generally branch several times, creating a new network to feed the tumor.
Do we completely understand this process? Well, no; for example, we have little understanding of why the sprouts branch. Jeremy Gunawardena pointed out a very nice modeling paper from a couple of years ago (Bauer et al. 2007. A cell-based model exhibiting branching and anastomosis during tumor-induced angiogenesis. Biophys. J. 92 3105-21) that used cell-based modeling to offer some interesting insights about the mechanisms that may be responsible for branching. Alas, as far as I can tell from Google Scholar, this paper has only ever been cited by other modeling papers, although the question of what controls branching (and issues like the role of the cytoskeleton in branching) are active areas of research.
In this paper Bauer et al. set out to develop the first cellular-level model of how sprouts form and grow. Previous models treated the behavior of the entire sprout as being determined by the behavior of the single cell at the tip, instead of considering the different cells in the forming blood vessel as distinct entities that each make their own decision about what to do next. Not surprisingly, if you treat the rest of the cells as being slavish followers of the tip cell this makes it hard to see where branching comes from: these models can only simulate branching by forcing it to happen, i.e. by including a rule that says that branching happens at a certain site with a certain probability. The authors point out that the process of angiogenesis naturally falls into three different time and length scales: what happens inside an individual cell, how a cell behaves, and what effects the cell has on the environment around it. In this paper they model only the cell and environment levels, leaving the explicit modeling of intracellular signaling for a later date.
To understand how the cells in the sprout interact with each other and with the extracellular matrix, the authors put together a cellular Potts model that allows individual cells to behave independently. A cell may excrete proteases to digest matrix structures, gain traction by binding to matrix proteins, and move through the surrounding environment, depending on its local circumstances. If the VEGF concentration is high enough, and the cell is at the right point in its personal cell cycle, the cell may also decide to divide. (I should note that not everyone likes this approach to modeling: not all of the decisions the model cell makes can be soundly based on molecular-level information. On the other hand, the cellular Potts model offers the ability to integrate experimental data at many scales — molecular, cellular, tissue-level — and lots of people find that attractive. In this case the authors seem to have worked hard to inject biological realism into their model, so even if you hate Potts models I suggest you suspend disbelief, for now.)
A key feature of this model is that the structure of the extracellular matrix and the effects that the invading sprout cells have on it are explicitly modeled. The matrix is not homogeneous to start with (even before the cells have started to invade), so it’s easier to move through in some directions than others. It also includes tissue-specific cells, which in later models may carry interesting cell-surface molecules, but in this model are simply objects in the matrix that are hard to push out of the way. Thus the local environment of an individual cell consists of many factors: the pattern of matrix fibers, locations of tissue cells, and the local concentration gradient of VEGF, etc. The rules for cellular behavior lead each cell to prefer to move towards the VEGF source, to make strong adhesive interactions with the matrix, and to have a round rather than elongated shape. And of course, they prefer to stay connected to each other. The interplay between these different preferences is treated as an energy minimization problem. At each time step, the cell starts off in a particular position in a lattice and is asked whether it would like to move to a neighboring position. If the move would reduce overall “energy”, the cell moves; if not, it may still move with a probability dependent on the energy difference (the probability of moving is lower if the difference between where it is and the new position is large).
The authors wanted to use their model to address two major questions: (1) What does the VEGF gradient have to look like to explain observed sprout morphology? and (2) Does the structure of the extracellular matrix matter? The first question is important because there are several isoforms of VEGF, some of which are soluble and others that bind to matrix. Soluble VEGF would create a shallow, relatively uniform gradient; although individual cells would take up VEGF via receptors on the cell surface, they would not take up enough to make a difference in the overall concentration profile. Matrix-bound VEGF would be released from the matrix by the action of proteases secreted by the cells, but, unlike soluble VEGF, it would not be quickly replenished by diffusion. So in this situation the VEGF taken up by a cell would create a very steep concentration gradient in the local environment. Which of these two scenarios leads to sprout morphologies like those seen in vivo? It turns out that a soluble-VEGF-only gradient leads to fat, slow-moving sprouts, while a matrix-VEGF-only gradient leads to skinny, agile ones that look more real. One interesting feature is that even though the rules given to the cells include “prefer to be round”, the combination of chemotaxis and adhesion to the matrix generates elongated cell morphologies, as seen in real-life sprouts. Indeed, there’s evidence from in vivo experiments that if tumors are engineered such that they only produce soluble VEGF, the blood vessels they induce are much fatter, and fewer in number, than when the matrix-bound isoforms are expressed. Bauer et al.’s model implies that this change in morphology results from two factors: the shallower local gradients around the sprout lead to less effective chemotaxis by the tip cell, so the whole sprout moves more slowly; and, each individual cell has access to an essentially infinite supply of diffusing VEGF, so many cells can become activated to divide. This leads to the fatness.
The structure of the extracellular matrix became intensely interesting to the authors when they realized that their model recapitulated branching. In this model, then (this is worth emphasizing), branching comes directly out of modeling the behavior of individual cells as independent entities, responding to their local environment. What factors in the environment encourage branching? Analyzing movies of their simulations, they find that their sprouts frequently change direction to take the path of least resistance through the matrix, while still pushing in the general direction of the VEGF gradient. When there are two low-resistance paths, the tip cell of the original sprout may take one of them and the daughter of a proliferating cell further back in the sprout may choose to take the other. Calling these paths low resistance is a bit misleading, though, because these are not necessarily just holes in the matrix; strong adhesion to either the matrix or one of the tissue cells embedded in the matrix can also create the forces that lead to the formation of a new branch. The key result is that branching (in this model) is entirely dependent on the fact that the medium the sprout is migrating through isn’t homogeneous. Another interesting feature is that if branches of the sprouts bump into each other they can merge to form a loop, which means that the model also recapitulates the phenomenon called anastomosis. Both branching and anastomosis happen because they result in lower-energy states in the model; in other words, these larger-scale behaviors of the system (“emergent properties”) come directly out of simulating the chemical and mechanical dynamics that happen at the level of individual cells.
Bear in mind, though, that the fact that a model produces phenomena similar to those we see in vivo doesn’t necessarily mean anything. We lack ways of distinguishing the models that seem to fit for spurious reasons from the models that are telling us something profound. People seem to think that making a specific prediction and testing it is the gold standard for non-spuriousness, and certainly I find it hard to think of something better. But there may be many spurious ways of making an accurate prediction. I recently posted on a paper from the Paulsson lab that showed that you can accurately predict the effect of increasing gene expression on fluctuations by starting from two completely different theories of where the fluctuations might be coming from. Worrying about spuriosity may be an inevitable part of modeling in biology. Nevertheless, this model seems very interesting to me because it offers a way to look at questions that are hard to get at experimentally, and one can hope that the results of the modeling would lead to better experimental design.
For that to happen, though, we need experimentalists to notice that the modeling has been done. If you know anyone working in this area, would you be kind enough to pass this link along? Maybe experimentalists aren’t interested in models; or maybe they just don’t read Biophysical Journal all that often. The matrix of science is itself pretty inhomogeneous, but perhaps it’s still sufficiently well-connected that we can put these alternative explanations to the test.
Bauer AL, Jackson TL, & Jiang Y (2007). A cell-based model exhibiting branching and anastomosis during tumor-induced angiogenesis. Biophysical journal, 92 (9), 3105-21 PMID: 17277180
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