By Reddi A.

Bone morphogenetic proteins (BMPs) are osteotro-phic and osteoinductive cytokines. BMPs are pleio-tropic cytokines with activities not just on bone and cartilage but additionally on mind, tooth, pores and skin, middle, lung, kidney, and a bunch of alternative tissues. They act on chemotaxis. mitosis, differentiation, telephone survival and cellphone loss of life. The organic activities of BMPs are med-iated by way of binding to express BMP receptors forms I and II. there's a collaboration among the receptors and downstream substrates similar to receptor-regulated R-Smads 1, five, and eight. The phosphorylated R-Smads accomplice with Co-Smad. Smad4. and input the nucleus to start up the transcription of BMP-responsive genes. Smad6 inhibits the job of style I BMP receptor protein kinase. hence, the BMP receptors supply a finely regulated homeostatic approach with exams and balances.

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This involves expansion in powers of Again this method has severe limitations. Alternative methods are becoming available (Langford 1977) for pursuing bifurcating solutions away from u = uc ' LOGISTIC GROWTH OF A SINGLE SPECIES 4a Discrete Lag To obtain a model in which the methods of Chapter 3 as well as those of Chapter 2 can be displayed, I shall look at a model even simpler than those discussed at the end of Chapter 2. A popular instantaneous model for the popul- ation growth of a single species, distributed homogeneously in space and having a finite upper limit to growth, is the logistic equation dx rx(l-x/K) .

The model can be shown by Y-i3-yX, Choosing the first of these, the general nature of plotting the curve :! = 0 and the line :! = 0 as in Figure 3. I shall examine a particular version of this model, which keeps (45b) unchanged, and replaces (45a) by dX dt dX dt ox[{Y-B-y~r. il=Y + yij, Y>i3, (45c) i3. (45d) Y < Again the form below threshold is chosen to fit smoothly; take a pure exponential decay. one might just as well The equilibrium points are now (0,0), + + (X ,Y ) and (46) 34 To ensure that X+, X are real the parameters must satisfy It is readily verified that x+, X- are greater than ~ and y+, Y- are greater than a.

Ex-s. is o. (40) This equilibrium point is always stable. If x(t) is interpreted as the biomass of the organism there can be no lag, because of energy conservation. However if x(t) is interpreted as the number of organisms, then lag may be introduced as a crude way of allowing for the relation between the growth of individual single cell organisms and their likelihood of splitting. Caperon (1969) and Thingstad and Langeland (1974) discuss the dynamics of the chemostat in terms of number of organisms, and examine the effects of lag.

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