Creencias, recuerdos y sentimientos en las memorias de los estudiantes

5 .2 .1 M a th e m a tic a l r e p r e se n ta tio n

W hen a cytokine binds to its receptor, it initiates a series of biochemical events in the target cell th a t ultim ately causes a functional response like protein synthesis. This effect can be m athem atically represented by a function E : R+ [0, oo) which relates the cytokine dose to the degree of response it produces in its target. If the response has a finite upper bound, it can be normalised to a point in the interval [0,1]. The dose is usually measured as the quantity of ligand bound to receptors (for example, in Seymour and Henderson, 2001) or simply the local ligand concentration (for example, in Schweitzer and Anderson, 1992; Dale et a/., 1996; Chan et a/., 1999). Response m ust also be defined and can be any qiiantifiably observable effect of cytokine-receptor binding. This includes changes to levels of downstream proteins, enzymes or gene sequences activated by the receptor, changes to the levels of proteins synthesised as a result of the signal, and changes in the cell's level of activation (such as receptor density). The response may also be measured on a physiological or systemic (as opposed to molecular) level. For instance, Dinarello et

a/. (1987) measured response to lL-1 in rabbits in term s of changes to body tem perature. Three forms of dose response commonly used to model biochemical system s are linear, median effect and threshold functions (see Figure 5.1). The biological mechanisms behind each function are discussed in turn.

E{c) Jg(c) E{c]

c c

( c )

Figure 5.1: Dose response curves for (a) linear, (b) median effect, and (c) threshold dose response function.

5 .2 .2 L in e a r r e s p o n s e

Consider a dose response function which relates the num ber of receptor binding events to the concentration of a protein secreted in response. In Section 2.4.3, we discussed how receptor binding events are carried downstream via a m ulti-step intracellular signalling cascade which ultim ately results in protein synthesis. Normally, the original signal is am­ plified at every step of the cascade, so one receptor binding event activates many enzyme molecules, which cause the production of even more second messengers and so forth. This phenomenon, known as magnitude amplification, allows one receptor binding event to cause many molecules of protein to be synthesised (Koshland et a/., 1982; A lberts et al,

2002, p. 869-870). However, unless the signalling cascade feeds back upon itself, the num­ ber of protein molecules produced will be proportional to the num ber of receptor binding events. Hence the dose response function is linear, although its constant of proportionality may be extremely large. Figure 5.1(a) shows a typical linear dose response function.

5 .2 .3 M e d ia n e ffe c t r e s p o n s e

The intracellular signalling cascade triggered by a receptor binding event is usually not a linear sequence. A receptor often contains several functional domains, each of which can trigger a separate signalling pathway. These pathways can form networks th a t interact with each other or feed back upon themselves (Brugge and McCormick, 1999). Such networks can be difficult to analyse mathematically, but a family of functions called median effect functions are often a good empirical fit to data. They have the following general form:

E m a x E j

i + ( ^ y

m = (5.1)

where Emax and Emin are the responses to a maximal and minimal dose respectively. Also,

E D50 is the level of cytokine binding corresponding to a response halfway between Emax

and Emin^ while s is the slope coefficient.

Note th a t Emax = = 0 gives a standardised dose response function with range [0,1]. W hen s = 1, this is called a Michaelis-Menten function and it produces a rect­ angular hyperbolic dose-response curve; hence the response is said to show hyperbolic sensitivity (Koshland et al, 1982). W hen the dose is transform ed to a log scale, the graph is sigmoidal. This graph (with logarithmic dose and arithm etic response scale) is called a

semi-logarithmic plot. As s increases above 1, the dose-response curve becomes increas­ ingly switch-like or threshold. Initially, the stimulus produces little response, bu t once the dose passes the critical threshold E D ^q the response reaches its maximum value rapidly. The response curve is sigmoidal, and steeply sigmoidal on a semi-logarithmic plot; the response is said to be ultrasensitive. See Figure 5.1(b) for an example of an ultrasensitive Michaelis-Menten function.

A median effect relationship can describe a mechanism th a t amplifies itself up to a point, after which the amplification process is curtailed by the shortage of a rate limiting species. For instance, as (2.5) shows, the relationship between ligand concentration and proportion of receptors bound to ligand at equilibrium is a Michaelis-Menten function. Since it is usually difficult to relate the ultim ate effect of receptor action to proportion of receptors bound to ligand in a mechanistic way, an empirical model called th e Occupation Theory of ligand-receptor interaction is often used (see Kenakin, 1997, pp. 12-17). This says th a t response to ligand binding is directly proportional to the proportion of receptors occupied by ligand. Since the proportion of occupied receptors is itself a hyperbolic function of local ligand concentration (see (2.5)), response will be a hyperbolic function of ligand concentration.

According to the original theory by Michaelis and M enten (1913, quoted in Rubinow and Segel, 1980), when an enzyme facilitates the conversion of a substrate to a product via the formation of an interm ediate enzym e-substrate complex, the rate at which substrate is converted into product is a Michaelis-Menten function. Hence this form of response may be appropriate for a receptor with intrinsic enzyme activity (like receptor tyrosine kinases) or which is able to associate to enzymes which activate when the receptor binds to ligand (like receptors in the cytokine receptor superfamily) (Hill, 1998; A lberts et al, 2002, pp. 871-892). For instance, Bhalla and Iyengar (1999) used Michaelis-Menten kinetics to model enzymatic reactions in a range of intracellular signalling pathways.

5 .2 .4 T h r e s h o ld r e s p o n s e

A threshold response is the limit of a median effect response as the slope s becomes infinitely steep. A threshold response is undetectable below a critical threshold b u t jum ps to its maximum value once the threshold is achieved (see Figure 5.1(c)).

A threshold response is biologically vital in cellular processes such as division, differen­ tiation and repair. These require events to occur in sequence, so the next stage of the

process does not begin until the previous one has been completed. The cell must be able to ignore small perturbations due to stochastic effects b ut still respond decisively to a major stimulus (Ferrell, 1996). So it is unsurprising th a t several processes have been observed which show very steep threshold responses to stim ulation. They include IL-2 directed T lymphocyte mitosis (Cantrell and Smith, 1984), T lym phocyte production of IFN- 7 in response to occupation of T-lym phocyte receptors (Viola and Lanzavecchia, 1996) and IFN- 7 production in response to antigen by cytotoxic T lym phocytes (Slifka et uA, 1999). Also, Wang et a/. (1994) observed th a t IL-1/? mRNA must exceed a threshold before IL- 1 protein synthesis occurs.

A median effect function with a very high slope coefficient s approxim ates a threshold function. Several signalling mechanisms have been proposed which give high values of

s. They include substrates th a t are reversibly converted between two different forms by separate enzymes (Goldbeter and Koshland, 1981) and proteins which simultenously release an activator and an inhibitor th a t compete for absorption into the same type of binding site (Testorf et a/., 2001).

Another signalling mechanism th a t produces a steep response is the mitogen activated protein kinase (MAPK) family of signal transduction cascades, one of the m ajor mecha­ nisms th a t relays signals from cytokine receptors to the nucleus and cytoplasm (reviewed in Arbabi and Maier, 2002). For instance, a member of the M APK family called p38 medi­ ates production of monocyte cytokines in response to LPS and T N F -a (Foey et aL, 1998; M anthey et al, 1998). MAPKs have also been identified in many of the pathways stim u­ lated by LPS (reviewed in G uha and Mackman, 2001), as well as the signalling pathways of inflamm atory cytokines involved in RA (Schett et al, 2000).

A threshold response can also occur in certain types of kinetic systems th a t feed back upon themselves (Lewis et al, 1977). As dose is increased the system may undergo a bifurcation th a t creates a new stable equilibrium. Once th a t point is passed, the system quickly moves to the new equilibrium.

5 .2 .5 J o in t r e s p o n s e f u n c t io n s

Gells are usually exposed to a cocktail of different cytokines and other mediators. In fact, very few biological responses are mediated by a single cytokine (Nicola, 1994, p. 1). As a result, cytokines usually act in combination with other cytokines and mediators th a t can amplify or dam pen the response.

If n agents with dose levels ci, C2, . . . , c„ act simultaneously, then th e combined response is measured by a m ultivariate joint dose response function E (c i, C2, . . . , c„). E is a (usu­ ally unknown) function of the marginal dose response functions for each individual agent E i(c i), £2(0 2). The form of E is ultim ately governed by th e way in which th e signal

transduction pathways of each agent interact. In practice, the form of E has usually been determ ined phenomenologically (by its ability to predict experim ental observations) al­ though recently several mechanistic models of intracellular signal transduction networks have been published (for example, Bhalla and Iyengar, 1999, 2001). However, grouping together kinetic models for even a few of these pathways would produce a model th a t is prohibitively complicated. Hence several authors (Bhalla and Iyengar, 2001; Asthagiri and Lauffenburger, 2000) have suggested a modular approach. Once each signalling pathway has been kinetically modelled, the results of such a model can be captured in a response function or even a logic function (such as AND, OR, NAND) th a t can be combined with other pathways.

The effect an agent has can be classified according to the direction of its regulatory effect. Given a region A C and two dose vectors c = ( c i , . . . , c^_i, c^, c^_pi,. . . , c^}, c' = {ci , . . . , cj, Q + i , . . . , Cn}, we say th a t agent i is:

1. o f z e ro effect over A if E{c) = E{c') for any c, c' G A.

2. u p - r e g u la to r y over A if E{c) > E(c') whenever c > c' and c, c' G A.

3. d o w n - r e g u la to r y over A if E{c) < E(c') whenever c > c' and c, c' G A.

Note th a t in principle an agent can exhibit both up-regulatory and down-regulatory effects, depending on the local concentration regime. For instance, T N F -a can cause neutrophilia at low doses and neutropenia at high doses (van der Poll et a/., 1992). Similarly, IL-10 can have both suppressive and enhancing effects on the immune system depending on the timing, dose and location of its expression (Moore et al, 2001).

5.3

R eview of literature about T N F -a , IL-1 and IL-10 in