/* * Mathematical Model for the Early Development of the Sea Urchin * Embryo * * Model Status * * This model has been edited to include initial concentrations * and correct dimensional errors. The model runs in COR and PCEnv * to produce oscillating graphs but has not been determined to * exactly replicate the experimental results. * * ValidateCellML verifies this model as valid CellML with consistent * units. * * Model Structure * * During normal mitotic cell division, eukaryotic cells will replicate * their DNA during the S-phase of the cell cycle, and then divide * during the M-phase. S and M phases are temporally separated * by gaps, called the G1 and G2 phases respectively. However, * in early development, when the main aim of embryo is to increase * cell number by a series of rapid cell divisions, the cell cycle * has a number of unique features: * * interdivision time is short and nearly constant; * * S and M phases alternate in time without intervening G1 or G2 * phases; * * cell division is nearly synchronous and most cells take part; * * and the nucleocytoplasmic ration (NCR) of cells increases dramatically. * * It is believed that the NCR controls the cell cycle characteristics * during the early stages of devlopment. This idea has been incorporated * into the following mathematical model of early devlopmental * cell cycle control. * * The molecular components involved in mitotic control have been * studied extensively in the last decade, and a control system, * common to all eukaryotes, has been elucidated. This has been * the subject of several mathematical models, including models * which use budding yeast as the experimental organism: * * Modelling the control of DNA replication in fission yeast, Novak * and Tyson, 1997 * * Mathematical Model of the Fission Yeast Cell Cycle, Novak et * al., 1998 * * Chen et al., Modelling the Budding Yeast Cell Cycle, 2000 * * Ciliberto et al. Modelling the Morphogenesis Checkpoint in the * Budding Yeast Cell Cycle, 2003 * * and those which use the Xenopus oocyte and embryo: Ciliberto * et al. A Kinetic Model of the Cyclin E/Cdk2 Developmental Timer * in Xenopus laevis embryos, 2003. * * In the Ciliberto and Tyson 2000 publication described here, * the authors develop a mathematical model based on the molecular * interactions underlying early embryonic cell cycle control (see * below). By introducing the NCP ratio into the molecular mechanism, * through model simulations the authors were able to reproduce * many of the physiological features of early sea urchin development. * Since the cell cycle controls appear to be common to all eukaryotes, * this model can also potentially be applied to other species. * * The model has been described here in CellML (the raw CellML * description of the Ciliberto and Tyson 2000 model can be downloaded * in various formats as described in ). * * The complete original paper reference is cited below: * * Mathematical Model for Early Development of the Sae Urchin Embryo, * Andrea Ciliberto and John J. Tyson, 2000, Bulletin of Mathematical * Biology , 62, 37-59. (A PDF version of the article is available * to subscribers on the Bulletin of Mathematical Biology website.) * PubMed ID: 10824420 * * reaction diagram * * [[Image file: ciliberto_2000.png]] * * The M phase control system of the cell cycle. The upper part * of the figure shows the two positive feedback loops involving * active MPF (Cdc2-cyclin complex) and inactive MPF (the phosphorylated * Cdc2-cyclin complex). Cyclin is produced at a constant rate * from a pool of amino acids (AA) and immediately binds to free * Cdc2, producing active MPF. All forms of cyclin - free, active * and inactive MPF - are degraded by the same process. The lower * part of the figure shows that the degradation of cyclin is the * result of a negative feedback loop involving a ubiquitin-ligation * complex (APC) and a putative intermediary enzyme (IE). */ import nsrunit; unit conversion on; unit minute=60 second^1; unit first_order_rate_constant=.01666667 second^(-1); math main { realDomain time minute; time.min=0; extern time.max; extern time.delta; real Y(time) dimensionless; when(time=time.min) Y=0.01577552; real k1 first_order_rate_constant; k1=0.024; real k2(time) first_order_rate_constant; real k3 first_order_rate_constant; k3=1.15; real C(time) dimensionless; real M(time) dimensionless; when(time=time.min) M=0.0142077; real kwee(time) first_order_rate_constant; real kcdc25(time) first_order_rate_constant; real preMPF(time) dimensionless; when(time=time.min) preMPF=0.02541723; real Cdc25P(time) dimensionless; when(time=time.min) Cdc25P=0.4844362; real k25 first_order_rate_constant; k25=18; real Km25 dimensionless; Km25=0.1; real k25r first_order_rate_constant; k25r=0.8; real k25ro first_order_rate_constant; real Km25r dimensionless; Km25r=1; real p dimensionless; real Cdc25(time) dimensionless; real Wee1(time) dimensionless; when(time=time.min) Wee1=0.5155638; real kw first_order_rate_constant; kw=18; real Kmw dimensionless; Kmw=0.1; real kwr first_order_rate_constant; kwr=0.8; real kwro first_order_rate_constant; real Kmwr dimensionless; Kmwr=1; real Wee1P(time) dimensionless; real IEP(time) dimensionless; when(time=time.min) IEP=0.002287817; real kie first_order_rate_constant; kie=4.5; real Kmie dimensionless; Kmie=0.01; real kier first_order_rate_constant; kier=0.34; real Kmier dimensionless; Kmier=0.01; real IE(time) dimensionless; real APC_(time) dimensionless; when(time=time.min) APC_=0.5051103; real kap first_order_rate_constant; kap=0.3; real Kmap dimensionless; Kmap=0.01; real kapr first_order_rate_constant; kapr=0.3; real Kmapr dimensionless; Kmapr=1; real APC(time) dimensionless; real Cdc2tot dimensionless; Cdc2tot=1; real Cdc25tot dimensionless; Cdc25tot=1; real Wee1tot dimensionless; Wee1tot=1; real IEtot dimensionless; IEtot=1; real APCtot dimensionless; APCtot=1; real R dimensionless; R=1; real s dimensionless; s=0.021; real q dimensionless; q=1; real V2 first_order_rate_constant; V2=0.01; real V2_ first_order_rate_constant; V2_=0.6; real V25 first_order_rate_constant; V25=0.04; real V25_ first_order_rate_constant; V25_=0.4; real Vwee first_order_rate_constant; Vwee=0.025; real Vwee_ first_order_rate_constant; Vwee_=2.5; // // Y:time=(k1-(k2*Y+k3*Y*C)); // M:time=(kcdc25*preMPF+k3*Y*C-(k2*M+kwee*M)); // preMPF:time=(kwee*M-(k2*preMPF+kcdc25*preMPF)); // Cdc25P:time=(k25*M*Cdc25/(Km25+Cdc25)-k25r*Cdc25P/(Km25r+Cdc25P)); k25ro=(k25r/p); // Wee1:time=((-1)*kw*M*Wee1/(Kmw+Wee1)+kwr*Wee1P/(Kmwr+Wee1P)); kwro=(kwr/p); // IEP:time=(kie*M*IE/(Kmie+IE)-kier*IEP/(Kmier+IEP)); // APC_:time=(kap*IEP*APC/(Kmap+APC)-kapr*APC_/(Kmapr+APC_)); // C=(Cdc2tot-(M+preMPF)); // Cdc25=(Cdc25tot-Cdc25P); // Wee1P=(Wee1tot-Wee1); // IE=(IEtot-IEP); // APC=(APCtot-APC_); // p=(s*R+q); // kcdc25=(V25*Cdc25+V25_*Cdc25P); k2=(V2*APC+V2_*APC_); kwee=(Vwee*Wee1P+Vwee_*Wee1); }