Speaker: Louis Vermeulen

Department: JNI Oncology

Subject: Stem cell dynamics in homeostasis and cancer of the gut

Location: Erasmus MC

Date: 23-3-16

 Author: Hielke Walinga

 

A good way to look to cancer in the biological sense is to see it as a process of evolution on the cellular level. A cancer cell is fitter than the healthy cells because it produces more offspring, and is not killed by apoptosis. In this way cancer can better be understood by studying the dynamics between cells and so discover what is responsible for a larger offspring in the cells. The cells that produce offspring in an organism are exclusively the stem cells. Studying stem cell dynamics is therefore very important in a better understanding of cancer. This is what Louis Vermeulen discussed in his seminar on 23 March. Louis Vermeulen has studied the stem cell dynamics of the intestinal stem cells (ISC’s) and proposed a model describing these dynamics and how certain cancer mutations alter these dynamics.

First of all research has shown that colon cancer isn’t a bulk of cells growing very fast, but it looked like these cancer create a bit their own small organized organs. This is, however, simply explained by the fact that these cancers didn’t arise from random cells, but did arise from stem cells. Therefore they grow still in a bit organized manner.

The inside of the intestine consists of a lot of relief, because it is covered with so-called villi. The cells in these villi got lost pretty fast and of course need to be replaced. The stem cells replacing these are located at the bottom between these villi. Replacing is therefore from the bottom to the top. In this well the ISC’s are located in a circle. Not only did these ISC’s replace the cells above them, but it turned out they also replace each other. This is discovered by the use lineage tracing. The stem cells are marked by something which is also visible in their offspring.

After this discovery they tried to model this stochastic dance of replacing. They discovered that their model was sufficient by taking only two parameters, N and λ. N is the number of ISC who participate in the dance and λ is the replacement rate of ISC’s per time unit. It turned out that the dance is so dynamic that there is a good chance that one ISC will replace all the others. This is called fixation, and obviously the fixation chance is the important value which a cancer cell is trying to increase.

When all ISC’s are healthy (i.e. no cancer cells) the chance that one ISC will replace its neighbor is just 50 percent. A cancer cell will of course have a higher chance to do this. When this chance is known the fixation chance can be calculated by the following formula.
Fixatie formule
Image 1: Formula for chance of fixation, Pfix, with the amount of ISC, N, and the increased chance of replacing its neighbor, PR (for neutral drift this is 0.5).

Later research has shown that certain cancer mutations indeed have a bias in this drift. These mutations are: KRASG120, APC+/-, APC-/-. Important to know is the a very frequent cancer mutation, p53R172N, is not creating a bias.

I think this research perfectly shows how mathematical models can give more insight in something biological complex as oncogenesis.

Fixatie evenement
Image 2: This image shows how a fixation event takes place.

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Exploring the roads to colon cancer

JNI Oncology Lectures

Speaker: Louis Vermeulen (Academic Medical Center Amsterdam)

Subject: Stem cell dynamics in homeostasis and cancer of the gut

Location: Erasmus MC Rotterdam

Date: Wednesday, 23.03.2016, 11:00-12:00

Author: Edgar Schönfeld

The inner lining of the  intestine has a surface area of circa 250 m2 and is completely renewed every 7 to 10 days. The high turnover rate might partially explain why colorectal cancer is one of the most common types of cancer. New tissue is continuously generated by intestinal stem cells (ISCs) that are located at the bottom of the intestinal crypts. However, the ISC population is not very static. For instance, ISCs can transition between a highly proliferative and a quiescent phenotype. Of particular importance here is the fact, that ISCs are constantly replaced by neighboring ISCs. The dynamics of this ‘birth-and-death’ process can be visualized by a technique called ‘lineage tracing’. Thereby an ISCs is labeled via the incorporation of a marker, such as a fluorescent protein, which will be expressed by the progenitors of the cell. Figure 1 shows the arrangement of ISCs in a cross-section of the bottom of an intestinal crypt. Imagine an initial situation in which each of these cells is labeled with a different marker. Over time, each ISC will either replace a neighbor or will be replaced itself. The result of this stochastic process is that one ISC will eventually replace all the other cells with its copies (this is called ‘fixation’).

ISCs

Figure 1: A schematic representation of the gradual replacement of ISCs by neighboring cells, leading to the fixation of one ISC phenotype.

The cells of origin of colorectal cancer are known to have stem-cell-like properties. With this in mind, Vermeulen and colleagues established a connection between ISC dynamics and the development of colorectal cancer. The dynamics of ISC birth and replacement are governed by several parameters, such as N and λ, representing the number of functional stem cells and the replacement rate measured in stem cells/day, respectively. Vermeulen et al. determined N to be 5 and λ to be 0.1 in the small intestine [1]. Under normal conditions, each ISC has a chance of 0.5 of replacing a neighbor, and a fixation-chance of 1/N. Hereby, we speak of ‘neutral drift’. Subsequently, the researchers went on to examine the effect of mutations in the APC tumor suppressor gene and the K-ras oncogene on the dynamics of this system. Of note, these two genes play a central role in the so-called adenoma-carcinoma sequence, a model stating among others that loss of APC must occur before activation of K-ras in order to induce colorectal cancer. Tumorigenesis is thought to be an evolutionary multistep process, in which the cell population with the greatest selective advantage will eventually dominate in the tumor. Indeed, Vermeulen and colleagues found that an APC or K-ras mutation endows ISCs with a selective advantage with regard to the observed replacement dynamics: While a normal ISC has a chance of 50% of replacing a neighbor, the chance rises up to 79% for APC negative and to 78% for K-ras positive ISCs. In addition, they found that loss of TP53, a gene that is mutated in 50% of all human cancers, does not alter the chance of replacing a neighbor, except in the case of colitis (inflammation of the colon lining). The chance of fixation can be calculated as follows: Pfix=(1-(1-0.PR)/PR)/(1-((1-PR)/PR)N), resulting in a fixation chance of ca 73% for a K-ras or APC mutation (where N=5 and PR= chance of replacing a neighboring cell).
As a consequence of this research, we should look out for drugs that can restore the balance in favor of normal ISCs.

 

[1] Vermeulen L, Morrissey E, van der Heijden M, Nicholson AM, Sottoriva A, Buczacki S, Kemp R, Tavaré S, Winton DJ. Defining stem cell dynamics in models of intestinal tumor initiation. Science. 2013 Nov 22;342(6161):995-8.

 

The Push and Pull of Mitochondrial Gymnastics

Speaker: Benoit Kornmann

Department: School of and biochemistry, Georgia Tech Department

Subject: The Push and Pull of Mitochondrial Gymnastics    

Location: Department of Bionanoscience, TU-Delft

Date: 17-03-2016

Author: Mirte Golverdingen

    

Benoit Kornmann works in a Cell Biology lab which is interested in dynamics on organelles. Their research mostly focusses on the mitochondria.

He started his talk with a simple picture of the cell. However, this picture is not how a real life cell is like. There are a lot of organelles in one cytoplasm and therefore the cytoplasm is very crowded. The organelles are completely intertwined in a compact environment. Furthermore, the organelles in the cell move, this means that there is a dynamic environment in the cell.

Kornmann focusses mainly on the mitochondrial dynamics. Mitochondria can be transported through the cell along cytoskeletal tracks. They have also the ability to fuse together and to fission apart. All these actions need mechanical forces to do their work. Kornmann focusses on the origin and the effects of these mechanical forces.

Mitochondrial transport is based on microtubules transport. It is promoted by Miro, kinesin and dynein. Miro interacts with the mitochondria and is in contact with the kinesin and the dynein. Kinesin is a motor protein that moves to the + end of the microtubules. While the motor protein dynein moves to the – end of the microtubules.

Kornmann’s research group showed that miro interacts with the protein Cenp-F. This protein is expressed in the nucleus during the S, G2 and the mitosis phase of the cell cycle. During mitosis Cenp-F first moves to the nuclear envelope following by the movement to the kinetochore. During the telophase and cytokinesis Cenp-F is requited to the mitochondria. We can also see this by using the CRISPR-Cas technique. If you silence miro by using this technique, the Cenp-F is not distributed to the mitochondria.

Cenp-F is a very large protein and it is made of a lot of coiled-coils. It has two microtubules binding domains at the N and C terminus of the protein. However, there is no binding site for motor proteins, so what then is the mechanical force that results in the movement of mitochondria? Recent research showed that the growing of microtubules can also do mechanical work. Therefore, the research group of Kornmann asked the question: does Cenp-F harness this force?

If Cenp-f does use the mitochondria to create mechanical force it needs to be located at the tip of the microtubules. Kornmann showed that the protein indeed is located on the tip of the microtubules. He did this by using super-resolution microscopy and immune-fluorescence. Eb1 was one of the labeled proteins and it is located on the microtubule. Cenp-F also labeled with another color. The images from the microscope showed that Cenp-F indeed was found at the top of the microtubules. Kornmann also used TIRF microscopy to confirm his claim.

Honours 3
Figure 1: Miro Recruits Cenp-F to mitochondria. Miro is necessary for Cenp-F mitochondrial recruitment. Localisation of Cenp-F (green) and mitochondria (Red) in the presence and absence of Miro1, Miro2 or both. Top: Cenp-F signal only. Middle: overlay. Bottom: magnification of the boxed areas. R: Pearson coefficient colocalisation analysis ± SEM. Scale bar, 10 μm. 2-4 regions were selected per cell. Number of selected regions: Miro1CRISPR: 98; siMiro2: 36 ; Miro1CRISPR-siMiro2: 34. Adapted from: Kanfer, G., Courthéoux, T., Peterka, M., Meier, S., Soste, M., Melnik, A., … Kornmann, B. (2015). Mitotic redistribution of the mitochondrial network by Miro and Cenp-F. Nature Communications, 6, 8015. http://doi.org/10.1038/ncomms9015

 

Kornmann concluded from these result that that miro is interacting with the Cenp-F protein. Cenp-F is then interacting with the tip of the microtubules. It is a very simple idea that justifies the results. They also reconstructed an in vitro system by using a stable microtubule seed on a plate. On this they put growing dynamic microtubules. They also added a glass bead that was coated with Cenp-F. They used microscopy to follow the bead and they saw the growth and shrinkage of the bead. This also confirmed the hypothesis that Cenp-F binds to the tip of the microtubules.

The network of mitochondria in the cell is in constant movement. How do the organelles avoid entangling with each other? Kornmann did research to this question by using the bacteria Shingella flexerni Once this bacterium is in a cell it high jacks the actin system, so they can infect the neighboring cell. When a bacteria bump into a mitochondrion, the mitochondrion will undergo fission. It is a biochemical process where Drp1 plays also a great role. The Drp1 protein makes a ring that activates the fission and splits the cell. They controlled if Drp1 is needed for this fission in the cell and it indeed was. When Drp1 in the mitochondria was inactivated by using the CRISPR-CAS method the mitochondria could not undergo fission. However, they can undergo fusion

How is the bacterium sensed? How does Drp1 know that it assembled right where the bacterium hits the mitochondria? Kornmann thought this could be sensed by a change in the mechanical forces on the Drp1 molecule. So the mitochondria feel that they are squeezes. By using Atomic Force Microscopy with a round tip they could put a force on a very flat cell. By applying a force across the cell membrane they were able to cause mitochondrial fission. Could this also be done on a more natural way than ATM?

For this question they let cells grow on a patterned surface. In your body your cells also do not grow on a flat plane. They used the edge of a groove on a vinyl gramophone record to grow cells on. They looked how the separation of mitochondria on the edge was organized for the wild type and for Drp1 CRISPR-CAS and Drp1 siRNA silenced cells. When the cell did not have the Drp1 protein the mitochondrion did not undergo the fission process. However, the wild type did undergo fission on the edge of the groove. Furthermore, every mitochondrion that went over the edge did undergo the fission process. Kornmann also showed that the mechanical force on the mitochondria fission site recruits Drp1 to this site. So fission can take place.

During this talk it became again clear that a cell is much more complex than you can imagine. The cytoplasm is very crowded and, therefore, the mitochondria have developed a mechanism to be able to move through the cell without being damaged. I did not know before that mitochondria cells can undergo fission. Moreover, I never realized that they need the microtubules to travel through the cell. It is exciting that the Kornmann group could show us in a simple way by using interesting new techniques as CRISPR-CAS how microtubules are able to travel through the crowded cytoplasm.

The push and pull of mitochondrial gymnastics: mechanical force in mitochondrial dynamics.

Speaker: Benoît Kornmann
Department: Bionanoscience
Subject: The push and pull of mitochondrial gymnastics: mechanical force in mitochondrial dynamics.
Location: Room E – Building 22 (TU Delft)
Date: 17-03-2016

Author: Kristian Blom

At the 17th of March, I visited a Bionanoscience seminar given by Benoît Kornmann, group leader of the Kornmann Lab of ETH Zurich. The Kornmann lab studies the ultrastructural organization of the cell and the biology of organelles.

Prof. dr. Kornmann introduced the main topic of the seminar with a drawing of the internal structure of an eukaryotic cell. Such pictures are typically used in textbooks to get some more insight in the anatomy of the cell, but they are rather misleading. The missing factor in most textbook images is the entropy and dynamical behavior of the internal structure. During mitosis, this chaotic internal environment needs to be regulated to make sure that both daughter cells get approximately the same cellular content (e.g. cytoplasmic organelles). But the molecular mechanism that allow proper segregation of cytoplasmic organelles in human cells are poorly understood. The research group of Dr. Kornmann focused on the mitotic redistribution of mitochondria during mitosis.
First I shall summarize the conclusion of the research, after that I will tell something about the methods used to obtain the results of the research.

During mitosis, mitochondria move towards the equatorial plate before partitioning into the daughter cells. This movement is established by microtubule-based transport. The bond formed between the mitochondria and the microtubules is established by the proteins Miro and Cenp-F. Miro, a mitochondrial protein, is located at the outer membrane of the mitochondria. During cytokinesis, Miro recruits Cenp-F, which associates with the plus-end (growing tip) of growing microtubules. This protein-microtubule interaction ensures the mitochondrial transport towards the equatorial plate, and the redistribution of the mitochondria over the two daughter cells.

The Miro-Cenp-F interaction observed by the research group of Dr. Kornmann, came as a surprise. At first sight, the main focus was to identify Miro interactors at mitochondria. While analyzing the Miro complex, a lot of unrelated proteins were identified. So, to get rid of these unrelated proteins, they applied a strategy called SILAC: Stable isotope labeling by amino acids in culture. After this, only three proteins were observed: Miro1, Miro2 (together they form Miro) and Cenp-F. Together with statistical analyses, this observation strongly suggests that Miro forms a complex with Cenp-F.

Image 1: Localization of Cenp-F (Green) and mitochondria (Red) in the presence and absence of Miro1, Miro2 or both. Top: Cenp-F signal only. Middle: Overlay. Bottom: magnification of the boxed areas. Source: Mitotic redistribution of the mitochondrial network by Miro and Cenp-F, 2015, Benoît Kornmann.

 

To know whether Miro is the causing agent for the recruitment of Cenp-F to the mitochondria, the Kornmann Lab used complete loss-of-function of both Miro1 and Miro2. Using the CRISPR/Cas9 method, Miro1 was mutated. By knocking down Miro2 in the same cell line, Miro-less cells were created. This complete silencing of the Miro protein led to the disappearance of Cenp-F at the mitochondria, no matter which cell cycle the cells were in. At the same time, Cenp-F was expressed in other locations of the cell (nucleus, nuclear envelope, etc.), suggesting that Miro is necessary for Cenp-F recruitment. To show that Miro is not only necessary, but also sufficient for Cenp-F recruitment, they overexpressed Miro1 and Miro2. This overexpression led to increased Cenp-F recruitment at the mitochondria.

This blog is too short to discuss all the different methods used for this research, but still I would like to give a remark about the second method which I did explain. By overexpressing Miro1/2 and observing an increase of Cenp-F recruitment at the mitochondria, you can’t conclude that Miro is sufficient for Cenp-F recruitment from my point of view. Because, there could be some protein or other chemical agent which is needed above a certain threshold before Cenp-F recruitment becomes activated. The concentration of this hypothetical chemical agent doesn’t influence the increased Cenp-F recruitment when Miro1/2 is overexpressed, but it’s still necessary for Cenp-F recruitment.

 

 

3D mammalian genome organization and function

Speaker: Wouter de Laat

Department: Bionanoscience

Subject: 3D mammalian genome organization and function          

Location: TU Delft

Date: 11-2-16

Author: Hielke Walinga

Our DNA consists of long strands of bases that appear to have no function at all. Only 3 % of the DNA is actually consisting genes. A long time researchers thought only the genes of the DNA mattered. However, it becomes more and more clear that the junk DNA hides a lot of information related to the expressing of these genes. Not only did they turn out to consist of a large landscape of regulatory elements, they also contain parts which guide to fold the DNA in a specific 3D construction which is also influencing the expressing pattern. Wouter de Laat investigated this 3D construction and talked about methods to reveal this, but also talked about a mechanism explaining the self-organization of this 3D construction.

When taking into account that junk DNA surrounding the genes are important to the expressing of the genes, one can deduce that the position of these genes in the DNA is important for the expressing pattern. A good way to test this is to place genes at random positions in the DNA. The Spitz lab executed these kinds of experiments and indeed showed that the expressing patterns can sometimes change very much.

A way to explain this is given by the hypothesis that when DNA strands are looping, the gene and its enhancer are placed close to each other in space. To test this hypothesis, a technique called Chromosomes Conformation Capture (referred to as 3C) is developed. This technique makes use of formaldehyde proteins which connect to the DNA, creating hairballs of DNA. These hairballs are the loops of DNA. When this is digested, the remaining DNA is all in small parts which used to be the loop. In the next step, it uses PCR with primers linking to the genes and its enhancer. The PCR then reveals if the gene and its enhancer were located in the same loop.

Other more elaborate techniques of the previous mentioned technique are the 4C and the Hi-C method. In the 4C method the topological distance between one locus is measured to all other DNA. The Hi-C method on the other hand measures all loci to all other loci. This method reveals beautifully how DNA is topologically organized.

Next, Wouter de Laat wanted to explain us about the construction of these loops. His hypothesis stated that on the root of the loop there are two CTCF anchors recruiting condensin. This condensin links the strands together creating the loop. To prover this hypothesis, researchers removed one anchor and observed that no loop is formed. (This was then tested with the 3C technique.) Another experiment showed a more surprising result. When one anchor was switched the loop was also not created.

At first, this made sense, but at the end of the presentation somebody made an important note. In a three dimensional world, switching one anchor does not automatically result in a prevention of linking. Especially when the loop is very large, this would make no difference. An important feature Wouter de Laat missed in his presentation was the coiling and super-coiling of the DNA. He missed to mention and elaborate on the physics of DNA. A hole in biology Nanobiology is hoping to fill.

Hi-C
Image 1: A heat map showing the topological distances in chromosome 14. This was created by the Hi-C technique. (Source: Lieberman-Aiden E, van Berkum NL, Williams L, Imakaev M, Ragoczy T, Telling A, Amit I, Lajoie BR, Sabo PJ, Dorschner MO, Sandstrom R, Bernstein B, Bender MA, Groudine M, Gnirke A, Stamatoyannopoulos J, Mirny LA, Lander ES, Dekker J,Comprehensive mapping of long-range interactions reveals folding principles of the human genome, Science 326, 289–293, 2009)

 

 

The beauty and the beast – cryo-electron tomographic reconstruction of two enveloped viruses

Speaker:      Sai Li

Department: Bionanoscience

Subject:       The beauty and the beast – cryo-electron tomographic reconstruction of two enveloped viruses.

Location:     TU Delft

Date:            03-03-2016

 Author:        Katja Slangewal

The spikes on the surface of a virus, even if you buy the most expensive light microscope, you won’t be able to see them. Sai Li from Oxford University told us what you can do instead. As the title of his seminar already reveals, there are ‘beautiful’ viruses and, as he calls them, ‘beasty’ viruses. Sai Li used the ‘beautiful’ Tula virus to explain how you can reconstruct the spikes and he used the ‘beasty’ Lassa virus to show what these reconstructions can tell you.

The envelopes of viruses can be visualized by using cryo-electron tomography and subtomogram averaging. The main idea of this process is to combine x-ray crystallization and cryo-EM imaging. See table 1 for the advantages and weaknesses of both methods.

Table 1: Advantages and weaknesses of crystallization and Cryo-EM

Crystal structure by X-rays Cryo-EM
+ High resolution + sample preparation, complete, in situ
– sample preparation, partial structure, artifacts – Low resolution

First I will explain the process for the Tula virus (the beauty) and afterwards I will continue with the Lassa or ‘beasty’ virus. The first reason why the Tula virus is a beauty is because it is a non-pathogenic virus. The virus also lacks a matrix, the glycol protein spikes serve as a matrix instead. This makes it easier to retrieve the structure of the spikes. The spikes of the Tula virus exist of a pair of glycoproteins: GN and GC. GN is a tetramer which attaches to the surface of the virus and GC is a class-II fusion protein. GC enables the Tula virus to merge membranes with a host cell.

In order to visualize the spikes of the Tula virus, pictures of the frozen virus are taken using the Electron Microscope. These pictures are sorted in two groups: containing spikes and not containing spikes. An average of the ones with spikes is taken to model the surface of the Tula virus and to form a 3D density chart of the virus. Sai Li found 3 regions on the surface that contain spikes. He called it an elegant spike interconnection, because it was EM friendly.

Than the crystal structure of GN of the Puumala virus (which is 80% identical to the Tula GN) was used to find the best mathematical fitting of the protein in the density chart formed before. At first the third domain of GC did not seem to fit. However this domain is connected by a flexible linker, which is hard to crystalize. By using some clever tricks Sai Li managed to proof that also this third domain does fit into the density chart. In the end a well conserved area of the spike was clearly exposed. This indicates an antibody binding site, which is useful for virus research.

Secondly the Lassa virus was discussed. The Lassa virus is ‘beasty’ because it is dangerous for humans and there is no treatment yet. Also this virus has a defense mechanism which produces fake viruses with almost the same features, which makes it difficult to purify the virus. It took Sai Li and his group a year to purify this virus.

The surface of the Lassa virus was reconstructed in the same way as the Tula surface, although there were more difficulties with this ‘beasty’ virus. In contrary to the EM friendly spike distribution of the Tula virus, the distribution of spikes on the Lassa virus is totally random. This means that information of neighbors can’t be used by reconstructing the shape of the spikes.  Another ‘beasty’ characteristic was the fusing pH of the Lassa virus. Normally viruses fuse at a pH of 4-5, the Lassa virus fuses at a pH of 3-4. Finally the Lassa virus can’t fuse when the protein Lamp1 is absent. The question is when the spikes do enable fusion and how important the pH and the Lamp1 protein are. By using cryo-electron tomography Sai Li found the answers.

Sai Li used virus-like particles without a genome to avoid danger. He looked at the spikes while changing the pH to see what happened with the volume of the spikes. See table 2 for the results of this experiment.

Table 2: volume changing after changing the pH

pH 7 5 3
Volume 100% 88% 46%

Sai Li also made combined images of the spikes in different pH, like he did with the Tula virus. Apparently the spikes open at a pH of 5 and they lose subpart GP1, which explains the difference in volume. There already was a supposed binding site of Lamp1, based on the amino-acid composition of both proteins. During the research it became clear that this binding site is only available after opening of the spike (see figure 1). This information is important in order to find treatment for this ‘beasty’ virus.

Lassa virus spike at different pH

Figure 1: The Lassa spike opening at pH 5, at pH 3 the GP1 subpart is lost. http://journals.plos.org/plospathogens/article?id=10.1371/journal.ppat.1005418

I liked the talk of Sai Li, it really showed how physics and biology are combined to Nanobiology. I was able to follow most of what Sai Li told and the ‘red line’ was totally clear. Sometimes I found it hard to understand the technical terms, especially during the part about the method. However, he used really clear images and movies which made it much easier to understand the story. I also really liked that he had a sense of humor. I learned a lot during this seminar. I already knew much about viruses and a bit about EM and crystalizing proteins. This seminar brought these three together in a nice way. I find it interesting to learn more about viruses and how to visualize them, although I am not sure I would like to study this myself. It sounded like a lot of programming and physics and only a biological context. Nevertheless Sai Li gave a really nice talk.

 

 

Identification of slowly reacting variables in a dynamic system using the Wasserstein metric

Speaker: Prof. Dr. Sjoerd Verduyn Lunel
Department:
Applied Analysis
Subject:
Identification of slowly reacting variables in a dynamic system using the Wasserstein metric
Location:
Utrecht University
Date: 2016-0
3-08
Author: Romano van Genderen

Of the seminars given during the Dutch National Mathematics Symposium I visited, I chose to share the seminar given by professor Verduyn Lunel with you. I chose this particular professor because I have already once visited one of his lectures on real single-variable analysis, and this topic because it has the most practical applications.

He first started slowly with the basics. He explained that a time series is a series of measurements of the same thing, done once every time interval t. A time series can be generated through measurement of a physical phenomenon or by a simpler model function. As an example, which was also recently mentioned in Computational Science, he used the Newton-Raphson method. This formula defines a variable x_{n+1} based on its previous value x_{n}. This also happens in another well-known formula, namely Hénon’s equation.

Next, he explained about attractors. When you have a time series, it has an attractor. This is a point or set of points A, where if the time series gets in the vicinity of A,in a set of points called V, it will never leave V. In the case of the Newton-Ralphson method, it is the root of the function, in the case of Hénon’s equation, a complicated 2-dimensional shape. Every time series has an attractor.

But the question he asked, can you predict the attractor when you have a time series. Kennel et al (1992) showed that this is possible by grouping specific terms in the time series into vectors and adding a lag between the terms. For example, the time series (1, 2, 3, 4, 5, 6) can be grouped like (1, 4), (2, 5), (3, 6). These are vectors in R^2, but sometimes other dimensions or lags are required.

So now you have a set of vectors {v_1, … v_n} ∈ R^n. Next, you should put the distance between these vectors inside a matrix. So now you have the matrix:

ea89f30c06c85680bb3b42f4f44add32.png

You can plot the columns of these matrices in a vector space. This leads to a point cloud. The fact that you have transformed a time series into a point cloud allows the most innovative mathematical object in this seminar to be used, the Wasserstein metric. This metric is a specific way to assign “distance” or “difference” between two point clouds. The way to visualise the Wasserstein distance is to think of all points in a point cloud as small heaps of sand. In that case, the Wasserstein distance is the least amount of work to be done on the heaps in the configuration of the first point cloud to push them into the second configuration. If these point clouds are very similar, you need just a little work to push the heaps of sand. But if they differ a lot, a lot of work is needed. So a low Wasserstein distance means a low difference between the two point clouds. The parameters in the model equation change the Wasserstein distance. And because the Wasserstein distance is related to the attractor as shown before, the attractor shifts position when the parameters change.

flow_eqw_opt.jpg

Fig 1. Image explaining the principle behind the Wasserstein metric, the gray heaps u are moved to the darker heaps w. Scott Cohen, Stanford University.

Now for all these things to come together. I showed that you can change a time series into a point cloud. You can calculate an objective distance or a measure of difference between two point clouds using the Wasserstein metric. So you can objectively say how different two time series are. This has many practical uses. But because prof. Verduyn Lunel was already a bit short in time, he only mentioned two.

The first is if you use a specific point in an MRI as your time series. In this case, you can objectively see the difference between two points in time. Instead of letting a doctor guess if something is severe or not, risking potential bias or human mistakes, you now have a simple number stating how much difference there is between a sick and a healthy patient. This also helps set a simple number for when intervention is needed.

The second use, which was a case study prof Verduyn Lunel participated in at the Academic Medical Centre in Amsterdam, was to distinguish between asthma and Chronic Obstructive Pulmonary Disease. Using time samples of patients breathing into what he called “Digital Noses”, he could distinguish between the patients having asthma and COPD, even noticing that one patient was suffering from both, a fact that could not have been observed earlier.