Friday, January 13, 2017

Microbe Makes the Shortest Path Problem Easy (But Not Really)

by Meghan Maltby
Physarum polycephalum
So, obviously all of us understand the shortest path problem common to computer science.  And obviously, we all know about the acelluar slime mold, Physarum polycephalum. And that these two things, an algorithmic solution living in the binary world and an organism living in the physical world, are incredibly and surprisingly connected.

But just in case you forgot about these two topics, I’ll refresh your memory.

Researchers claimed in 2000 that that an acellular slime mold, a yellow, sticky, tubular growth shown in the picture above, solved the shortest path problem, which asks the question, what is the quickest way to get from one point to another (1)? In the article published in Nature, P. polycephalum determined the shortest path between two food sources.  The importance of that feat lies in the background of the algorithmic solution to the shortest path problem, which I will explain first.  The cellular machinery behind the biological solution is incredible, and will be discussed after.

The shortest path problem lives within the realm of Graph Theory, which is the study of graphs. I know you’re thinking:
But in this case, you’d be wrong.  Graph theory, as it relates to computer science, is a little different than your standard algebra class where you use a set of axes to depict functions. In computer science, graphs look like this:


Where the numbered circles are called nodes and the lines are called edges.  The nodes are destinations and the edges are paths to travel between those destinations.  These graphs are not normally drawn to scale, so weights, represented by the letters, are added and associate a cost with moving along any given single or multiple edges.  This is similar to deciding to take the highway or side streets in rush hour.  The highway may be more direct, having a smaller number of edges to traverse, but taking side streets probably saves time even though the distance is longer.

The shortest path problem asks a simple question, if weights are additive, what is the shortest path between two nodes? There are different variations of this question, depending on what type of problem is being solved.  But for this example, let’s say we are trying to find the quickest path between nodes 1 and 6.

The first famous algorithm that tackled this problem was published by Edsger W. Dijkstra in 1959 (2). The solution is based on two factors; one, the weights of the edges, and two, which nodes are connected. The algorithm moves systematically through the graph, starting at node 1, and determines the placement of nodes and edges (2).  If the algorithm started at node 1, it would find two edges emanating from it; edge a to node 5 and edge b to node 2.  The algorithm will pick a newly found node, say node 2, then determine what edges and nodes it has.  This continues until the entire graph has been discovered.
From discovering node 1 and node 2   
It keeps track of which nodes it has visited and which nodes it hasn’t in a matrix by saving the weight of the edge that connects two nodes. If two nodes do not have a connection, the matrix saves a value of infinity (2). After inputting information gathered from nodes one and two, the matrix would look like this:


There is also another matrix that keeps the minimum weight of a path from one node to another.  This matrix is a bit more complicated. A truncated version is shown with possible pathways between node 1 and any other node.  What is shown in the result from the completion of the program, after the entire graph has been discovered. In reality, the matrix would compare values for different paths, and only save the one that was the shortest:


As you can see, the algorithm gets complicated quickly.  The bigger the graph, the bigger the matrices, and if you’re looking for the shortest distance between any two given nodes…phew.

Now the question is, did this slime mold actually solve the shortest path problem?

The answer is yes. Researchers created a maze and placed small pieces of P. polycephalum throughout and let it grow until the organism filled it entirely (1).  The figure below shows the maze after this point. Blue lines depict the possible ways the organism could grow to create a path between the two food sources (which are shown in the figure after this one, labeled AG), and are labeled by a1, a2, b1, and b2.

There are four possible paths through the maze between the food sources. The slime mold needed to decide which path to take at two different points; between a1 and a2, and between b1 and b2.  a2 was 22% shorter than a1, and b1 and b2 only had a 2% difference (1). Four hours after the food sources were placed, the maze looked like this, which shows that only the connections between the two food sources were kept by leaving growth in all possible pathway solutions, and any growth leading to dead ends was moved.

Noticing this, the researches left the organism for another four hours.  By maximizing the solution the organism chose pathway a2b1, after eight hours which was the best solution to the maze (1).

Researchers were thrilled, and called this incredible show of “foraging efficiency” a primitive form of intelligence. Foraging means obtaining food and nutrients, and completing it efficiently means the organisms fitness increases. Intuitively, taking the shortest path from the cave to the watering hole is the most efficient because it saves both time and energy.

So how does this slimy mold solve a complicated problem in a matter of hours?

It has to do with its physiology.  Because P. polycephalum is acellular, its body consists of one giant cell, so there are no walls separating one side of the organism from another.  That means that its contents, including water, nutrients, and proteins, move without being inhibited. But that does not mean that the movement is not controlled.  The slime mold pulsates, like the beating of a heart, by alternating the direction of water current (3).  The water carries with it Calcium ions, which cause the contraction of the plasma membrane. This is analogous to squeezing a water balloon or stress ball, where the force that you exert with your hand causes the object to bulge out.  This contraction and relaxation cycle moves the front edge of the plasmodium forward.

Here is a gif, adapted from “This Pulsating Slime Mold Comes in Peace (ft. It's Okay to Be Smart) | Deep Look” YouTube  video, of the cellular contents of the plasmodium sloshing back and forth:

P. polycephalum protoplasm movement (2)
make action GIFs like this at MakeaGif

The current within the cell can move in any direction, and changes based on the environment (3).  There are two stages to P. polycephalum movement.  The first is the searching phase, where the majority of the body mass is radiating outward in a fan shape in search of food (4). After the fan gets big enough, there is not enough body mass to cover the surface where the fan has been, so the organism creates veins.  Here is a snapshot from a video that shows four separate droplets of food where P. polycephalum is consuming the first and searching for more:

The veins create a network, which aids in movement and nutrient absorption. The network serves as communication channels between the body mass consuming the food drop and the body mass in search of new food.  Because there is limited protoplasm, most of the mass stays and consumes food, making absorption more efficient.  As the protoplasm searches, the network thins out, thickening veins that lead to new food sources by removing “legs” that lead to dead ends (3). The following snapshots from the same video show this in detail:
https://www.youtube.com/watch?v=mvBSkt6LhJE    
Once the protoplasm finds food, the organism enters the second stage of movement, called the feeding stage (4). Here, almost the entire body mass is moved to the location of the new food source, seen by it being covered completely with yellow slime.
When the food source runs out, the organism moves back to the searching stage, and fans out, creating veins until it finds its next meal:
The solution to the shortest path problem lies in the efficiency of the movement style.  As we saw, the protoplasm only leaves a few thick tubes between food sources. From hydrostatic theory, thicker tubes are more efficient than thin tubes at passing nutrients (3).  Think of it like this, if you needed to put out a house fire, you wouldn’t try to push water from a fire hydrant through a straw.  And along the same analogy, if you were putting out a house fire, you would use the fire hydrant that was closest, rather than running the fire hose down another block.  By having thick and short pipes, the organism solves the seemingly complex shortest path problem, simply by maximizing its efficiency.

So, now what?

We’ve learned about the algorithm that answers the question, what is the shortest distance between two places given a bunch of different paths?  And we’ve learned that life, compact in a slime mold, solves the exact same problem without storing tables of numbers.  Though the shortest path problem is an interesting intellectual endeavor for humans, the shortest path problem is life or death for this organism, and it seems to understand intuitively that the shortest path between two points maximizes its likelihood of surviving.  Because of this amazing feat, researchers study Physarum polycephalum to understand the fundamentals of what makes life, alive.



References
1.        Nakagaki T, Yamada H. 2000. Maze-solving by an amoeboid organism. Nature 407:470.
2.        Dijkstra EW. 1959. A note on two problems in connexion with graphs. Numerische Mathematik 1:269–271.
3.        Nakagaki T, Kobayashi R, Nishiura Y, Ueda T. 2004. Obtaining multiple separate food sources: behavioural intelligence in the Physarum plasmodium. Proceedings of the Royal Society B: Biological Sciences 271:2305–2310.
4.        Halvorsrud R, Giaever I, Feder J. 1997. Growth and Synchronization of Physarum Plasmodia on a Lattice of Disconnected Nutrient Drops. Biological Rhythm Research 28:358–364.


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