Molecular Topology Networks

In preparation for a grant I’m writing, I’ve been reading about graph theory and network topology. This has led me down multiple rabbit holes, some productive, some less so. Today’s productive reading comes from “Topology of molecular interaction networks” (Winterbach, W.; Van Mieghem, P.; Reinders, M.; Wang, H.; de Ridder, D. BMC Systems Biology 2013, 7:90). It’s an excellent overview of network biology, and very helpfully classifies extant research approaches as Descriptive analysis, Suggestive Analysis, and Predictive analysis. Each has its pros and cons, clearly laid out.

 

It’s nice to read a topology paper that itself is well-organized both topically and topologically. There are some excellent tables, with clear definitions. Here’s Table 1 on metric descriptions. I’ve been reading up on graph theory so most of the terminology was familiar, but I wish I had read this paper first because it would have eased my reading of several other dense papers, that were not always as well-organized.

 

To figure out what’s more closely related to what, we typically employ a distance metric. This is usually associated with a numerical value. Algorithms allow us to apply statistical measures, and out pop new numerical values – perhaps telling us something about how closely things are clustered, or whether there is an overall modular architecture, or whether a network is scale-free. We can compare different networks. We can compare one numerical value to a different numerical value.

 

I also found clear descriptions of how we can compare real-world networks to randomly simulated (null) networks, why it is useful, and helpful summary statements about different methods to do this. Even though I had come across this before, I hadn’t quite grasped the importance of the null network comparison until reading this paper. As I read I was jotting down useful language and terminology for my grant.

 

There’s a helpful paragraph where they describe two views of robustness, a concept I’ve been wrestling with. “One property thought to emerge from natural selection is robustness, the ability to maintain function under perturbations… [in some] networks, the number of interaction partners of nodes initially appeared to correlate with their essentiality: … [they] have few hubs and many low-degree nodes. In metabolic networks, almost the opposite is true, with networks being susceptible to disruption of low-degree linker nodes that connect metabolic modules. However, in both cases the systems are resilient to most perturbations but susceptible to targeted attacks, a property known as highly optimized tolerance.”

 

It’s always hard to read a paragraph pulled from a paper, but trust me if you do read this paper, what the authors are saying becomes quite clear. They also highlight limitations at every turn. In the next paragraph, they refer to simulations showing that “such structural features emerge from network dynamics rather than selective pressure.” This cleared up a lot of confusion for me. (I apologize if you find this confusing, but I write my blog to offload excess memory, and this is very helpful to me.)

 

But what if you don’t have a distance metric? I alluded to this in my most recent post because my challenging reading for the day was “The Topology of the Possible: Formal Spaces Underlying Patterns of Evolutionary Change” (Stadler, B. M. R.; Stadler, P. F.; Wagner, G.; Fontana, W. J. Theor. Biol. 2001, 213, 241-274). I’ve been playfully calling this “Nearness is Not a Number”.

 

At first glance, this seems like an impossible task. If you can’t measure the distance between A and B, and also between A and C, how can you tell which is more closely related to A? Is it B or C? And could you relate B and C somehow? Turns out you can still do this using set theory. This paper was a bear to read. I glossed over much of the math beyond the initial definitions. However, Figure 3 (shown below) in the paper was very helpful. I’m used to working in Euclidean vector space, and have started to think about metric space where I’m more limited in the mathematical operations I can use, but I can now see how there are other non-numerical operations or features you can use for your groupings. I also have a new appreciation for how fuzzy boundaries can be utilized. 

 

This blog post does not have a conclusion. I’m soaking in information; I don’t quite know what to do with some of it yet. Maybe when I sleep and dream, new connections will be made. Topology is a strange and funny beast to the non-mathematician me. I’m just beginning to glimpse it usefulness to interesting questions in systems chemistry.