Conservation Laws in Cellular Automata

Conservation laws in physics are numerical invariants of the dynamics of a system. In cellular automata (CA), a similar concept has already been defined and studied. To each local pattern of cell states a real value is associated, interpreted as the “energy” (or “mass”, or . . . ) of that pattern.The overall “energy” of a configuration is simply the sum of the energy of the local patterns appearing on different positions in the configuration. We have a conservation law for that energy, if the total energy of each configuration remains constant during the evolution of the CA. For a given conservation law, it is desirable to find microscopic explanations for the dynamics of the conserved energy in terms of flows of energy from one region toward another. Often, it happens that the energy values are from non-negative integers, and are interpreted as the number of “particles” distributed on a configuration. In such cases, it is conjectured that one can always provide a microscopic explanation for the conservation laws by prescribing rules for the local movement of the particles. The onedimensional case has already been solved by Fuk´s and Pivato. We extend this to two-dimensional cellular automata with radius-0,5 neighborhood on the square lattice. We then consider conservation laws in which the energy values are chosen from a commutative group or semigroup. In this case, the class of all conservation laws for a CA form a partially ordered hierarchy. We study the structure of this hierarchy and prove some basic facts about it. Although the local properties of this hierarchy (at least in the group-valued case) are tractable, its global properties turn out to be algorithmically inaccessible. In particular, we prove that it is undecidable whether this hierarchy is trivial (i.e., if the CA has any non-trivial conservation law at all) or unbounded. We point out some interconnections between the structure of this hierarchy and the dynamical properties of the CA. We show that positively expansive CA do not have non-trivial conservation laws. We also investigate a curious relationship between conservation laws and invariant Gibbs measures in reversible and surjective CA. Gibbs measures are known to coincide with the equilibrium states of a lattice system defined in terms of a Hamiltonian. For reversible cellular automata, each conserved quantity may play the role of a Hamiltonian, and provides a Gibbs measure (or a set of Gibbs measures, in case of phase multiplicity) that is invariant. Conversely, every invariant Gibbs measure provides a conservation law for the CA. For surjective CA, the former statement also follows (in a slightly different form) from the variational characterization of the Gibbs measures. For one-dimensional surjective CA, we show that each invariant Gibbs measure provides a conservation law. We also prove that surjective CA almost surely preserve the average information content per cell with respect to any probability measure.

To divide or differentiate? : nestin-mediated regulation of Cdk5 in cell fate decisions

The molecular functions of the non-cell cycle-related Cyclin-dependent kinase 5 (Cdk5) have been of primary interest within the neuroscience field, but novel undertakings are constantly emerging for the kinase in tissue homeostasis, as well as in diseases such as diabetes and cancer. Although Cdk5 activation is predominantly regulated by specific non-cyclin activator protein binding, additional mechanisms have proved to orchestrate Cdk5 signaling in cells. For example, the interaction between the intermediate filament protein nestin and Cdk5 has been proposed to determine cellular fate during neuronal apoptosis through nestin-dependent adjustment of the sensitive balance and turnover of Cdk5 activators. While nestin constitutes a crucial regulatory scaffold for appropriate Cdk5 activation in apoptosis, Cdk5 itself phosphorylates nestin with the consequence of filament reorganization in both neuronal progenitors and differentiating muscle cells. Interestingly, the two proteins are often found coexpressed in various tissues and cell types, proposing that nestin-mediated scaffolding of Cdk5 and its activators may be applicable to other tissue systems as well. In the literature, the molecular functions of nestin have remained in the shade, as it is mostly exploited as a marker protein for progenitor cells. In light of these studies, the aim of this thesis was to assess the importance of the nestin scaffold in regulation of Cdk5 actions in cell fate decisions. This thesis can be subdivided into two major projects: one that studied the nature of the Cdk5-nestin interplay in muscle, and one that assessed their role in prostate cancer. During differentiation of a myoblast cell line, the filament formation properties of nestin was found to be crucial in directing Cdk5 activity, with direct consequences on the process of differentiation. Also the genetic knockout of nestin was found to influence Cdk5 activity, although differentiation per se was not affected. Instead, the genetic ablation of nestin had broad consequences on muscle homeostasis and regeneration. While the nestin-mediated regulation of Cdk5 in muscle was found to act in multiple ways, the connection remained more elusive in cancer models. Cdk5 was, however, established as a significant determinant of prostate cancer proliferation; a behavior uncharacteristic for this differentiation-associated kinase. Through complex and simultaneous regulation of two major prostate cancer pathways, Cdk5 was placed upstream of both Akt kinase and the androgen receptor. Its action on proliferation was nonetheless mainly exerted through the Akt signaling pathway in various cancer models. In summary, this thesis contributed to the knowledge of Cdk5 regulation and functions in two atypical settings; proliferation (in a cancer framework) and muscle differentiation, which is a poorly understood model system in the Cdk5 field. This balance between proliferation and differentiation implemented by Cdk5 is ultimately regulated (where present) by the dynamics of the cytoskeletal nestin scaffold.

Eosinophilia is a favourable prognostic marker in oral squamous cell carcinoma

Suusyöpä on yleisin pään ja kaulan alueen pahanlaatuisista kasvaimista. Niistä yli 90 % on levyepiteelikarsinoomia. Koska suusyöpäpotilaan viisivuotisennuste on vain noin 50 %, on jatkuva tarve löytää keinoja ennustamaan potilaan selviytymistä ja ohjaamaan hoitoa. Suun levyepiteelikarsinooman ympäristössä havaitaan säännöllisesti eosinofiilejä. Eosinofiili on ihmisen immuunijärjestelmän erikoistunut solu, joiden määrä lisääntyy sekä tulehdusreaktioissa että syöpien läheisyydessä. Toistaiseksi ei tiedetä, miten eosinofiilit liittyvät suun levyepiteelikarsinoomaan, mutta oletuksena on, että suusyöpään liittyvällä eosinofilialla, TATE (tumor-associated tissue eosinophilia), voisi olla vaikutusta suun levyepiteelikarsinoomapotilaan ennusteeseen. Tämän tutkimuksen tarkoituksena oli selvittää TATE:n ilmenemistä ja vaikutusta potilaan ennusteeseen suun levyepiteelikarsinoomassa. Lisäksi tutkimus käsittelee potilaan ennusteen kannalta optimaalista eosinofiilimäärän raja-arvoa, jota voitaisiin käyttää patologin työkaluna ennusteen arvioinnissa. Tutkimusaineisto koostui Turun yliopiston Suupatologian laitoksella vuosina 2002-2010 tutkituista 122 suuontelon ja huulen limakalvokoepalasta, jotka oli otettu diagnostisia tarkoituksia varten 99 potilaalta. Tutkimuksen potilaista 44 oli naisia ja 55 miehiä, ja heidän keski-ikänsä oli 65,3 vuotta. Seuranta-aika oli keskimäärin 40,7 kk. Kaksi tutkijaa analysoivat hematoksyliini-eosiinilla värjätyt näytteet suurentamalla ne 400-kertaisiksi valomikroskoopilla. Eosinofiilien määrä laskettiin yhteensä kuudelta edustavimmalta syövän ja strooman alueelta. TATE:n suhde potilaan kliinispatologisiin piirteisiin ja selviytymiseen selvitettiin Turun yliopistollisen keskussairaalan potilastiedoistoista ja analysoitiin tilastollisesti käyttämällä Fischerin testiä. Työllä oli Varsinais-Suomen sairaanhoitopiirin eettisen toimikunnan lupa (nro T10/2011, päätös O31/11). Levyepiteelikarsinooman kliininen kuva vaihteli haavaisen muutoksen ollessa yleisin. Yleisin sijainti levyepiteelikarsinoomalle oli kieli. TATE:a löydettiin 61,5 %:sta (78/122) levyepiteelikarsinoomanäytteitä. Mikäli TATE:a ei löydetty tai sen määrä oli korkea, oli potilaan selviytyminen tilastollisesti merkitsevästi parempi kuin potilailla, joilla TATE oli matala. Lisäanalyyseissä havaittiin, että potilaan ennuste oli tilastollisesti merkitsevästi huonompi, mikäli TATE:n raja-arvo oli vähemmän kuin neljä eosinofiiliä per tutkittu mikroskooppinäkymä (HPF). TATE on täten merkki suun levyepiteelikarsinoomapotilaan paremmasta ennusteesta erityisesti, kun havaitaan enemmän kuin neljä eosinofiiliä/HPF. Tutkimustulosten varmistamiseksi tarvitaan kuitenkin jatkossa laajempia tutkimuksia.