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Wolfram Summer School

June 24–July 14, 2018
Bentley University, Waltham, MA


Enrique Barrajon

Class of 2011


Enrique Barrajon is a Spanish and European certificate in medical oncology. He holds a PhD in medicine from the Universidad de Navarra and a master's degree in statistics and research design from the Universidad Autònoma de Barcelona. He is Head of Medical Oncology at the Hospital Clinica Benidorm in Spain and a member of several national and international medical societies.

He is interested in mathematical modeling of cell growth, differentiation, mutagenesis, carcinogenesis, tumor development, immunology, clonal evolution, drug resistance, pharmacokinetics, and pharmakodynamics. Other interests are painting, mountain biking, and swimming.

Project: Substitution Systems Applied to the Growth and Differentiation of Caenorhabditis Elegans (C. Elegans)

C. elegans is a free-living soil nematode that is approximately 1 mm. in length. As the first metazoan to have had its genome sequenced, C. elegans has benefited greatly from both classical genetic and modern functional genomic approaches. Many human genes and pathways involved in cancer are highly conserved in C. elegans. Indeed, these regulatory networks are often easier to parse in C. elegans, as the gene families involved contain fewer members, thus reducing the likelihood for genetic redundancy.

Several characteristics of the worm make it highly amenable for cancer research. C. elegans has a completely characterized and essentially invariant somatic cell lineage, thereby facilitating the analysis of phenotypes that disrupt normal proliferation and patterning.

The embryonic cell lineage of Caenorhabditis elegans has been traced from zygote to newly hatched larva, with the result that the entire cell lineage of this organism is known. During embryogenesis, 671 cells are generated; in the hermaphrodite, 113 of these (in the male, 111) undergo programmed death, and the remainder either differentiate terminally or become postembryonic blast cells. The embryonic lineage is highly invariant, as are the fates of the cells to which it gives rise.

The complexity of a cell lineage can be defined as the length of its shortest algorithmic description, by analogy with Kolmogorov complexity, and is a function of three properties: the number of cell divisions that it contains, the number and distribution of cell fates that it gives rise to, and its topology or pattern of cell divisions. According to a model of cell complexity, C. elegans cells can be classified into 9 categories according to their morphology and function, based on standard anatomical descriptions: 39 blast, 113 death, 93 epithelial, 2 germ, 13 gland, 20 intestinal, 123 muscle, 46 neural structural cells, and 222 neurons.

Substitution systems allow the number of elements to change. In the simple cases, the rules specify that each element of a particular color should be replaced by a fixed block of new elements, independent of the colors of any neighboring elements. To get behavior that is more complicated than simple nesting, one must consider substitution systems whose rules depend not only on the color of a single element, but also on the color of at least one of its neighbors. If the rate of disappearances is too large, then almost any pattern will quickly die out, and if there are too few disappearances, then most patterns will grow very rapidly. But there is always a small fraction of rules in which the creation and destruction of elements is almost perfectly balanced. Allowing elements to various colors makes much more complicated behavior possible.

The aim of this project is to model the embryonic growth of C. elegans according to substitution rules with and without considering neighbor influence. The rule space will be searched for rules that allow a growth of elements in the range of hundreds, filtering for rules that produce extinction or an unlimited growth. Selected rules will be compared with the tree network of C. elegans.

Mutations in several genes associated with cell cycle control can induce somatic cell hyperproliferation defects in C. elegans. Such ectopic divisions, however, rarely produce anything resembling a classic tumor. Rather, supernumerary somatic cells undergo quiescence, terminally differentiate, and integrate more or less appropriately into their respective organs. By contrast, mutations leading to excess germline proliferation can result in bona fide tumors. Although generally confined to the gonad, germline tumor cells are mitotically active and fail to differentiate into gametes. In addition, as the only tissue to harbor a population of stem cells throughout adulthood, the C. elegans germline provides an opportunity to study mechanisms governing replicative immortality and the maintenance of pluripotency, both of which are relevant to traits acquired by cancer cells.

Stability tests will be performed looking for rule mutations able to produce extinction or hyperproliferation (unlimited growth), the hallmark of cancer, in some of the progeny representing the somatic and germline lineages of C. elegans.

If there is time, a model of hematopoietic growth and differentiation will be developed.

Project-Related Demonstrations

Growth in 1D Substitution Systems with Two Colors and Up to One Neighbor

View demonstration of Wolfram Demonstrations Project

Favorite Four-Color Totalistic Cellular Automaton

Rule 8584