Institute for System Dynamics and Control Engineering, University of Stuttgart, Pfaffenwaldring 9, 70569 Stuttgart, Germany

Max-Planck-lnstitute for Dynamics of Complex Technical Systems, Sandtorstr. 1, 39106 Magdeburg, Germany

Department of Pathology, Anatomy and Cell Biology, Thomas Jefferson University, 1020 Locust St., Philadelphia, PA 19107, USA

Abstract

Background:

Receptors and scaffold proteins possess a number of distinct domains and bind multiple partners. A common problem in modeling signaling systems arises from a combinatorial explosion of different states generated by feasible molecular species. The number of possible species grows exponentially with the number of different docking sites and can easily reach several millions. Models accounting for this combinatorial variety become impractical for many applications.

Results:

Our results show that under realistic assumptions on domain interactions, the dynamics of signaling pathways can be exactly described by reduced, hierarchically structured models. The method presented here provides a rigorous way to model a large class of signaling networks using macro-states (macroscopic quantities such as the levels of occupancy of the binding domains) instead of micro-states (concentrations of individual species). The method is described using generic multidomain proteins and is applied to the molecule LAT.

Conclusion:

The presented method is a systematic and powerful tool to derive reduced model structures describing the dynamics of multiprotein complex formation accurately.

Background

Receptor-mediated signal transduction is the subject of intense research since it plays a crucial role in the regulation of a variety of cellular functions. The ligand binding to a receptor triggers conformational changes that allow for receptor dimerization and phosphorylation of numerous residues. The subsequent formation of multiprotein signaling complexes on these receptors and their scaffolding adaptor proteins initiates a variety of signaling pathways. The number of feasible different multiprotein species grows exponentially with the number of binding domains, and can easily reach thousands or even millions

In 1998, the stochastic simulation tool StochSim was developed to handle the problem of combinatorial complexity

Recently, a new approach has been introduced where a mechanistic picture of all possible states is substituted by a macro-description that follows the occupancy levels and other characteristics of individual domains, e.g. the phosphorylation states of these sites

The contribution is structured as follows. First, we introduce our method using simple examples reconsidering some of the cases described in

Results

In order to introduce and discuss our method in an descriptive manner, we first want to introduce some definitions and outline the principles shortly. Afterwards, we will consider some simple generic examples (scaffold proteins with three and four binding domains) in more detail. Of course, these examples might also be treated without recourse to our theoretical developments, because the number of states is relatively small. But precisely because some of the results may appear intuitive, this shows the potential of our method which is also applicable to other more complicated problems. Additionally, we generalize two important cases to scaffolds with n binding sites (namely independent binding domains and domains that are controlled by one controlling site). We also provide the generalized transformation matrix for proteins with n binding sites and exemplify our method considering the scaffold molecule LAT (Linker of activation in T-cells).

Definitions

In the following, we will consider a receptor or scaffold protein _{i }different effector proteins _{i}, and hence can be in _{i }+ 1 different states (unoccupied or occupied by one of the _{i }effectors). The number of feasible micro-states for such a scaffold protein is

The number of possible reactions that have to be considered is

_{1}, 0, 0, 0] denotes all scaffold proteins

All these definitions shall be clarified by considering a simple scaffold protein _{1 }= 3 and _{2 }= 2 and _{1 }+ 1 = 4, namely either unoccupied or occupied by one of the three effectors. According to these considerations the number of different states of domain 2 is _{2 }+ 1 = 3. The total number of complexes results from ∏(_{i }+ 1) = 4·3 = 12. In order to calculate the number of reactions that may occur in this example we first assume that domain 1 is unoccupied. Now three different effectors may bind to this domain, which corresponds to three different reactions. At the same time binding domain 2 can be unoccupied or occupied by one of the two effectors

Reduction method

Our method can be divided into three essential steps. First, we start generating a complete mechanistic description of the considered scaffold or receptor like described in

**Step 1: **The reaction networks considered are described by a system of ordinary differential equations (ODEs). All feasible reactions _{on}_{A}_{B }- _{off}_{C }using the law of mass action. Here, _{on }and _{off }denote the association and dissociation constants, while _{i }refers to the concentrations of the components. The ODEs for all feasible micro-states have the form

In vector notation, Equation 4 can be written as

**Step 2: **In order to introduce new coordinates

In the following, we will introduce this transformation for a number of simple cases to exemplify our method and discuss the general pattern of this transformation. A strict mathematical and general formulation is also provided.

**Step 3: **Figure

State space transformation

**State space transformation**. The idea of our method can be described more easily by considering a mechanical example: In order to model the movement of a mass in space one has to choose a certain coordinate system. However, if this coordinate system is not adjusted to the problem (like shown on the left site) the model equations will be more complicate than in a transformed coordinate system.

Since ^{-1 }in order to retrieve the original variables

Generic example with three binding domains

We will start with a scaffold protein with three binding domains. In our example, each domain can bind one distinct effector molecule. Hence, the scaffold protein can exist in eight different micro-states (

Reactions for scaffold with 3 binding sites. A complete mechanistic model of a scaffold protein with 3 binding domains (1,2,3), where each domain can bind one effector protein (_{1}, _{2}, _{3}), has to consider the following 12 reactions. The kinetic parameters for each reaction can be denoted with _{+i }for the association and _{-i }for the dissociation reaction

Binding of _{1}

Binding of _{2}

Binding of _{3}

_{1 }⇋

_{2 }⇋

_{3 }⇋

_{1 }⇋

_{2 }⇋

_{3 }⇋

_{1 }⇋

_{2 }⇋

_{3 }⇋

_{1 }⇋

_{2 }⇋

_{3 }⇋

Functionality of a scaffold protein

Using the law of mass action and the reactions defined in Table

Domain interactions

**Domain interactions**. We assume that binding domain one controls the other two domains like indicated by the arrows. From this assumption the kinetic parameters for the model follow immediately. As soon as binding domain one is occupied, the affinities of the docking sites two and three will change. Since binding domain one is independent of the other binding sites, the on- and off-rate constants of this domain are also independent.

Hierarchical transformation

The second step after having formulated a complete mechanistic model is to perform a state space transformation. This transformation introduces new states, including the levels of occupancy of each domain. Choosing a globally invertible and smooth transformation assures that the system's dynamics is preserved and, as long as none of the new equations is eliminated, the original micro-states can be retrieved from the new ones at any time

We choose a hierarchical transformation matrix, consisting of different tiers (Table _{0 }will be constant. However, in the general case (including production and degradation) this is also an important dynamic and macroscopic quantity of interest. The 1st tier of our matrix corresponds to the discussed levels of occupancy of each binding domain (Equations 52b to 52d in our example). All states belonging to both of these tiers are called macro-states. The 2nd tier describes the levels of occupancy of all pairs of domains, corresponding to the concentration of scaffold proteins

State Space Transformation for scaffold with three binding domains. Transformation for a scaffold protein with 3 binding domains. The transformation is hierarchically structured and introduces macroscopic quantities like the overall concentration of _{0 }to _{3}), mesoscopic quantities describing pairs of concurrently occupied domains (_{4 }to _{6}) and microscopic quantities corresponding to individual multiprotein species (_{7}).

_{0 }=

_{1 }=

_{2 }=

_{3 }=

_{4 }=

_{5 }=

_{6 }=

_{7 }=

Independent binding domains

First, a reduced model describing the example with three independent binding domains shall be introduced. The mechanistic model (Equation 4) is transformed as specified in Table _{0 }is constant. Additionally, the output variables (levels of occupancy) which are described by the states _{1 }to _{3 }in the model appear to be completely decoupled from all other ODEs. This finding implies that a model describing the levels of occupancy does not need to consider the whole set of ODEs. They are

Transformed equations for independent domains. Transformed model equations for a scaffold protein with independent binding domains. The levels of occupancy (_{1 }to _{3}) do not depend on the states _{4 }to _{7}.

_{0 }= 0 (53a)

_{1 }= _{1 }(_{0 }- _{1}) _{1 }- _{-1}_{1 } (53b)

_{2 }= _{2 }(_{0 }- _{2}) _{2 }- _{-2}_{2 } (53c)

_{3 }= _{3 }(_{0 }- _{3}) _{3 }- _{-3}_{3 } (53d)

_{4 }= _{1 }(_{2 }- _{4}) _{1 }- _{-1}_{4 }+ _{2 }(_{1 }- _{4}) _{2 }- _{-2}_{4 } (53e)

_{5 }= _{1 }(_{3 }- _{5}) _{1 }- _{-1}_{5 }+ _{3 }(_{1 }- _{5}) _{3 }- _{-3}_{5 } (53f)

_{6 }= _{2 }(_{3 }- _{6}) _{2 }- _{-2}_{6 }+ _{3 }(_{2 }- _{6}) _{3 }- _{-3}_{6 } (53g)

_{7 }= _{1 }(_{6 }- _{7}) _{1 }- _{-1}_{7 }+ _{2 }(_{5 }- _{7}) _{2 }- _{-2}_{7 }+ _{3 }(_{4 }- _{7}) _{3 }- _{-3}_{7}. (53h)

One site controls the others

Now we assume that binding domain 1 controls the domains 2 and 3 (see Figure _{1 }to _{5 }are completely independent of states _{6 }and _{7 }(Table _{1 }to _{3}), the Equations for _{6 }and _{7 }can be excluded. Additionally, the Equations for _{1 }to _{5 }can be divided into three modules that are completely free of retroactive effects thus facilitating the analysis of the model as well as the application of parameter identification tools _{3}] and _{3}]) and _{3}] and _{3}]). However, this information is essential if binding domain 2 is controlled by domain 1, since in the first case the affinity of _{3}] to the effector _{3}].

Interaction motifs

**Interaction motifs**. Generic examples of scaffold proteins with 3 or 4 docking sites. (a) A scaffold protein with 3 distinct docking sites which do not interact. (b) Another pattern of domain interactions for the same scaffold protein. Here binding domain 1 controls the other two domains. (c) A scaffold protein with 4 docking sites and a more complex pattern of domain interactions. Each pattern of domain interactions can be translated into special kinetic properties (like exemplified in Figure 2).

Transformed ODEs for scaffold with on controlling domain. Transformed model equations for a scaffold protein with one controlling domain. The levels of occupancy (_{1 }to _{3}) are only influenced by the states _{4 }and _{5 }but not by _{6 }and _{7}.

_{0 }= 0 (54a)

_{1 }= _{1 }(_{0 }- _{1}) _{1 }- _{-1}_{1 } (54b)

_{2 }= _{2 }(_{0 }- _{1 }- _{2 }+ _{4}) _{2 }- _{-2 }(_{2 }- _{4}) + _{3 }(_{1 }- _{4}) _{2 }- _{-3}_{4 } (54c)

_{3 }= _{4 }(_{0 }- _{1 }- _{3 }+ _{5}) _{3 }- _{-4 }(_{3 }- _{5}) + _{5 }(_{1 }- _{5}) _{3 }- _{-5}_{5 } (54d)

_{4 }= _{1}_{1 }(_{2 }- _{4}) - _{-1}_{4 }+ _{3}_{2 }(_{1 }- _{4}) - _{-3}_{4 } (54e)

_{5 }= _{1}_{1 }(_{3 }- _{5}) - _{-1}_{5 }+ _{5}_{3 }(_{1 }- _{5}) - _{-5}_{5 } (54f)

_{6 }= _{2 }(_{3 }- _{5 }- _{6 }+ _{7}) _{2 }- _{-2 }(_{6 }- _{7}) + _{4 }(_{2 }- _{4 }- _{6 }+ _{7}) _{3 }- _{-4 }(_{6 }- _{7}) + _{3 }(_{5 }- _{7}) _{2 }+ _{5 }(_{4 }- _{7}) _{3 }- (_{-3 }+ _{-5}) _{7 } (54g)

_{7 }= _{1 }(_{6 }- _{7}) _{1 }- _{-1}_{7 }+ _{3 }(_{5 }- _{7}) _{3 }- _{-3}_{7 }+ _{5 }(_{4 }- _{7}) - _{-5}_{7 } (54h)

Analysis and dynamic simulations

In order to illustrate the advantages of our method, the previous example will be discussed (see Figure

Kinetic parameters for dynamic simulation

Affinity of domain

_{on }[^{-1}^{-1}]

_{off }[^{-1}]

Equilibrium _{d }[^{-1}]

1 (always)

_{1 }= 3·10^{5}

_{-1 }= 6

5·10^{4}

2 (domain 1 unoccupied)

_{2 }= 1

_{-2 }= 18

5.6·10^{-2}

2 (domain 1 occupied)

_{3 }= 5·10^{7}

_{-3 }= 24

2.1·10^{6}

3 (domain 1 unoccupied)

_{4 }= 1

_{-4 }= 12

8.3·10^{-2}

3 (domain 1 occupied)

_{5 }= 1·10^{5}

_{-5 }= 60

1.7·10^{3}

**Model 1: **We create a complete mechanistic model accounting for all molecular species and all possible reactions (compare Table

**Model 2: **We already mentioned that most heuristic models do not account for all molecular species. However, the equations of these models still describe the system at a microscopic level. The complete mechanistic network structure is substituted by a reduced structure focusing on a reduced number of species and a reduced number of reactions. As shown by Faeder et al. _{1}. Since the resulting affinity as well as the resulting on-rate of binding domain 2 is approximately several hundred-fold higher than the affinity or the on-rate of domain 3, we assume that the effector _{2 }in the majority of cases will bind before _{3}. Therefore, the reduced model only includes the following three reactions

_{1 }⇋

_{2 }⇋

_{3 }⇋

which are parametrized with the known kinetic constants. This case represents a commonly performed simplification. In _{1 }and _{1 }are in the state _{2 }and _{3}, and the structure of Model 2 does not allow _{1 }to dissociate after _{2 }and _{3 }have bound to the scaffold protein.

**Model 3: **At last we want to consider the reduced model, derived using our method (compare equations for _{1 }to _{5 }in Table _{1}, whereas the second consists of the two states _{2 }and _{4 }and the third of _{3 }and _{5}. The states of the second module do not influence the states of the third module and vice versa. The dynamics of _{1 }is neither influenced by the states of the second nor the third module, but _{1 }influences all the other states. Therefore, the model is not only modular but also hierarchically structured. This motivates a modular approach in order to analyze the model dynamics. First, one can analyze the dynamics of the first module. Afterwards the other two modules can also be analyzed separately. A similar approach is possible for the parameter identification since our transformation also leads to a modularization of the parameters. The only parameters of the first module are _{1 }and _{-1}. The parameters _{2}, _{-2}, _{3 }and _{-3 }can be found only in the second module, and the remaining parameters are present only in the third module. This allows one to identify the parameters step by step, which is a much simpler task than to identify all parameters at once. In addition, it also shows which parameters can be identified by certain measurements: for example, a measured time course of _{2 }(level of occupancy of domain 2) does not allow the identification of the parameters _{4}, _{-4}, _{5 }and _{-5}. The same also holds true for the complete model (model 1), but it is not intuitive to untangle this feature in the structure of the ODEs of the complete model. Because of all these advantages we think that the method proposed here offers a useful framework to handle multiprotein complex formation in signaling and regulation networks.

Dynamic simulations

**Dynamic simulations**. Dynamic simulations of the example shown in Figure 3b using the parameter values presented in Table 5. Here we compare the levels of occupancy of the three protein domains. The left picture shows the level of occupancy of domain 1, the second picture shows the levels of occupancy of domain 2 and the right picture shows that of domain 3. The results show that the reduced model we obtained by applying our method (model 3) accurately describes the real time course represented by a complete mechanistic model (model 1). The other model which follows from a number of reasonable simplifications which can also be found in literature provides completely different results.

Example with more than one controlling domain

In the previous chapter we analyzed scaffold proteins with completely independent binding domains or with only one controlling domain. However, our method can also be applied to a more general case. This will be exemplified by considering a scaffold protein with 4 binding domains, where each domain can be free or occupied by an effector. We assume that binding domain 1 controls domains 2, 3 and 4. Additionally, binding domain 3 also interacts with binding domain 4 (see Figure ^{4 }= 16. Applying our method to this example (details can be found in the Appendix), one finds that 9 equations are sufficient to describe the complete dynamics of the macro-states. The states that are required are the levels of occupancy of all 4 binding domains, the pairs of concurrently occupied binding domains {1,2}, {1,3}, {1,4} and {3,4}, as well as the triple of concurrently occupied binding domains {1,3,4}. This result shows that the more binding domains interact, more ODEs are required in order to describe the dynamics of the macro-states. Indeed, if all binding domains interact with each other, one really has to consider the whole combinatorial variety.

Scaffold proteins with n binding sites

Generalized transformation matrix

Here we want to formalize the transformation matrix for any scaffold protein with _{i }effectors compete for the binding domain _{i}. Hence, each binding domain _{i }+ 1 different states. The transformation matrix is independent of the intramolecular domain interactions. The 0th tier of our transformation matrix consists of only one state describing the overall concentration of the scaffold protein. The new state _{0 }corresponds to the sum of all feasible micro-states

The levels of occupancy of each binding domain are described in the 1st tier. The status of the binding domain _{i}} ) and one sums up all micro-states whose binding domain

with

with

The following tiers represent all possible tuples, and the last tier of the transformation matrix (the

with

Independent binding domains

Now we want to consider a scaffold protein with _{i }equations (i.e., states) instead of ∏ (_{i }+ 1) equations. A proof that in this case the macro-states are always sufficient to describe the system can also be found in

One site controls the others

We also want to generalize the case that binding domain 1 controls all other (_{i }- _{1 }instead of ∏ (_{i }+ 1).

Linker for activation of T cells (LAT)

LAT (Linker for Activation of T cells) is a scaffold molecule that plays a pivotal role in T cell signaling _{2}) and the generation of dyacilglycerol (DAG) and inositol trisphosphate IP_{3}. DAG activates RasGRP, which in turn activates Ras, as well as PKC, while IP_{3 }regulates Calcium signaling

PLC

Recent experimental data from LAT mutation studies indicate that the binding domains can influence one another, which contradicts the assumption of the complete independence

Discussion

We have presented an approach which allows one to create reduced models of multiprotein complex formation processes often occuring in signal transduction cascades, but also in regulation mechanisms (e.g. cell cycle regulation

Conclusion

We have shown that a complete mechanistic model of scaffold proteins or receptors as discussed in

Appendix

Proof

In order to prove that the transformation matrix is invertible, we will first show that the transformation matrix is quadratic, which is a necessary condition. Second, we will show that this matrix can be written as a upper triangular matrix, which is a sufficient condition for invertibility.

For each protein domain/site _{i}, 1, ..., _{i}}, where the numbers 1, ..., _{i }denote possible states of site _{i}replaces 0, which was used to designate the basal state of site _{i}). There are _{k }is straightforward: summing up all micro-states that (1) correspond to each character entry _{i }(from _{i}) and (2) have the states of the other domain given by the numbers in _{k}) equals the number of columns (the micro-states).

The next step is to proof that all columns of the transformation matrix are linearly independent by complete induction. If we look consecutively at all the new defined variables, starting with the last one in our listing above, one can show that each state is a sum of already defined states plus

Example with more than one controlling domain

In this example we assume that the considered scaffold molecule has four binding domains (1, 2, 3 and 4), which can bind four distinct effectors _{1}, _{2}, _{3 }and _{4}. The domain-domain interactions are depicted in Figure

_{0}

_{1 }= _{1}(_{0 }- _{1})_{1 }- _{-1}_{1 } (21)

_{2 }= _{2}(_{0 }- _{1 }- _{2 }+ _{5})_{2 }- _{-2}(_{2 }- _{5}) + _{3}(_{1 }- _{5})_{2 }- _{-3}_{5 } (22)

_{3 }= _{4}(_{0 }- _{1 }- _{3 }+ _{6})_{3 }- _{-4 }(_{3 }- _{6}) + _{5}(_{1}- _{6})_{3 }- _{-5}_{6 } (23)

_{4 }= _{6}(_{0 }- _{1 }+ _{10 }- _{13 }- _{3 }- _{4 }+ _{6 }+ _{7})_{4 }- _{-6}(_{4 }- _{7 }- _{10 }+ _{13}) (24)

_{7}(_{1 }- _{6 }- _{7 }+ _{13})_{4 }- _{-7}(_{7 }- _{13}) + _{8}(_{3 }- _{6 }- _{10 }+ _{13})_{4 }

- _{-8}(_{10 }- _{13}) + _{9}(_{6 }- _{13})_{4 }- _{-9}_{13 } (25)

_{5 }= _{1}(_{2 }- _{5})_{1 }- _{-1}_{5 }+ _{3}(_{1 }- _{5}) - _{-3}_{5 } (26)

_{6 }= _{1}(_{3 }- _{6})_{1 }- _{1}_{6 }+ _{5}(_{1 }- _{6}) - _{-5}_{6 } (27)

_{7 }= _{1}(_{4 }- _{7})_{1 }- _{-1}_{7 }+ _{9}(_{6 }- _{13})_{4 }- _{-9}_{13 } (28)

+ _{7}(_{1 }- _{6 }- _{7 }+ _{13})_{4 }- _{-7}(_{7 }- _{13})

_{10 }= _{4 }(_{4 }- _{7 }- _{10 }+ _{13})_{3 }- _{-4}(_{10 }_{13}) + _{5}(_{7 }- _{13})_{3 }- _{-5}_{13 } (29)

+ _{8}(_{3 }- _{6 }- _{10 }+ _{13})_{4 }- _{-8}(_{10 }_{13}) + _{9}(_{6 }- _{13})_{4 }- k_{-9}_{13}

_{13 }= _{1}(_{10 }- _{13})_{1 }- _{-1}_{13 }+ _{5}(_{7 }- _{13}) - _{-5}_{13 } (30)

+ _{9}(_{6 }- _{13})_{4 }- _{-9}_{13}.

In these differential equations _{0 }denotes the overall concentration of the scaffold protein. The states _{1 }to _{4 }represent the levels of occupancy of the four distinct binding domains. The remaining states describe the number of scaffold proteins with two or three concurrently occupied binding domains ( _{5 }corresponds to concurrently occupied binding domains {1,2}, _{6 }equals {1,3}, _{7 }{1,4}, _{10 }{3,4} and _{13 }{1,3,4}).

Linker for activation of T cells (LAT)

Applying our method to the adaptor protein LAT (Linker for activation of T cells), we derived two different reduced models (dependent of the assumptions that are made). First, we assume that the binding residues Y132, Y171, Y191 and Y226 are completely independent (see Figure

_{0 }= 0 (31)

_{1 }= _{1}(_{0 }- _{1})_{-1}_{1 } (32)

_{2 }= _{2}(_{0 }- _{2 }- _{3})_{-2}_{2 } (33)

_{3 }= _{5}(_{0 }- _{2 }- _{3})_{-5}_{3 } (34)

_{4 }= _{3}(_{0 }- _{4 }- _{5})_{-3}_{4 } (35)

_{5 }= _{6}(_{0 }- _{4 }- _{5})_{-6}_{5 } (36)

_{6 }= _{4}(_{0 }- _{6})_{-4}_{6}. (37)

The state _{0 }denotes the overall concentration of LAT molecules, which is assumed to be constant. The state _{1 }equals the level of occupancy of Y132 with PLC_{2 }and _{3 }describe the levels of occupancy of Y171 either with Grb2 or Gads, _{4 }and _{5 }are the equivalent values for the binding domain Y191, and _{6 }denotes the level of occupancy of Y226 with Grb2.

However, recent experimental results show that the binding domains of LAT influence each other. In a second example we assume that Y226 interacts with the two domains Y171 and Y191. The whole calculation can be found in the provided Mathematica-File. In this case the reduced model consists of ten ordinary differential equations, namely

_{1 }= _{1}(_{0 }- _{1})_{-1}_{1 } (38)

_{2 }= _{2}(_{0 }- _{2 }- _{3})_{-2}_{2 } (39)

_{3 }= _{5}(_{0 }- _{2 }- _{3 }- _{6 }+ _{14 }+ _{17})_{-5}(_{3 }- _{17}) (40)

+ _{6}(_{6 }- _{14 }- _{17})_{-6}z_{17}

_{4 }= _{3}(_{0 }- _{4 }- _{5})_{-3}_{4 } (41)

_{5 }= _{7}(_{0 }- _{4 }- _{5 }- _{6 }+ _{18 }+ _{19})_{-7}(_{5 }- _{19}) (42)

+ _{8}(_{6 }- _{18 }- _{19})_{-6}_{19}

_{6 }= _{4}(_{0 }- _{6})_{-4}_{6 } (43)

_{14 }= _{2}(_{6 }- _{14 }- _{17})_{-2}_{14 }+ _{4}(_{2 }- _{14})_{-4}z_{14 } (44)

_{17 }= _{4}(_{3 }- _{17})_{-4}_{17 }+ _{6}(_{6 }- _{14 }- _{17})_{-6}_{17 } (45)

_{18 }_{3}(_{6 }- _{18 }- _{19})_{-3}_{18 }+ _{4 }(_{4 }- _{18})_{-4}_{18 } (46)

_{19 }= _{4}(_{5 }- _{19})_{-4}_{19 }+ _{8}(_{6 }- _{18 }- _{19})_{-8}_{19}. (47)

(48)

Again the states _{0 }to _{6 }correspond to the same quantities as described above. The state _{14 }equals to all LAT molecules with Grb2 being concurrently bound to the residues Y171 and Y226, _{17 }describes the molecules with Gads being bound to Y171 and Grb2 bound to Y226. The states _{18 }and _{19 }are equivalent quantities describing the same binding motifs for Y191 and Y226.

Authors' contributions

HC conceived the original mathematical idea. HC, JSR and BNK developed the systematic method.

Model building and numerical analysis were carried out by HC, JSR and TS. EDG initiated, supervised and coordinated the project. All authors wrote the manuscript and approved the final version.

Linker for activation of T-cells

**Linker for activation of T-cells**. The four distal tyrosine rests on LAT and the binding possibilities, according to [26, 28].

Additional Files can be downloaded from

Click here for file

Acknowledgements

Special thanks to Michael Ederer for numerous and fruitful discussions, to Rebecca Hemenway for carefully reading the manuscript and for linguistical corrections and to Jon Lindquist for carefully reading the LAT chapter. HC, JSR, TS and EDG acknowledge support from the Deutsche Forschungsgemeinschaft (DFG), FOR521 and SFB495, and Bundesministerium fuer Bildung und Forschung (BMBF). BNK acknowledges support from the National Institute of Health Grant GM59570.