Mathematical modeling of intracellular signaling pathways

Abstract

Dynamic modeling and simulation of signal transduction pathways is an important topic in systems biology and is obtaining growing attention from researchers with experimental or theoretical background. Here we review attempts to analyze and model specific signaling systems. We review the structure of recurrent building blocks of signaling pathways and their integration into more comprehensive models, which enables the understanding of complex cellular processes. The variety of mechanisms found and modeling techniques used are illustrated with models of different signaling pathways. Focusing on the close interplay between experimental investigation of pathways and the mathematical representations of cellular dynamics, we discuss challenges and perspectives that emerge in studies of signaling systems.

Figure 1

Building blocks. Building blocks or modules of signaling pathways. Despite their diversity, signaling pathways often employ a set of common components (modules, motifs, building blocks) including the depicted ones. A) Receptors sense specific ligands or stimuli and change their conformation from the susceptible form R_s to the active form R_a, which transfers the signal downstream. Cells can fine-tune their excitability by (i) changing the susceptibility of receptors, e.g. switch between R_s and inactive form R_i, or (ii) by regulating the number of receptors via production or internalization and degradation. B) G proteins are heterotrimers consisting of subunits α, β, and γ. In the ground state, the α-subunit is bound to GDP. The active receptor triggers the activation of the G protein by exchanging the GDP with a GTP. The G protein dissociates into its subunits, which transmit the signal downstream by binding to other proteins and activating or inhibiting biochemical processes. The α-subunit carries the GTP. After GTP hydrolysis, which is a highly regulated process on its own, the subunits can reassociate to form the initial heterotrimeric G protein. The Regulator of G protein signaling (RGS) is involved in a larger feedback loop. C) Small G proteins switch between GDP-bound or GTP-bound forms with different activities. Conversion from the GDP state to the GTP state is catalyzed by a so-called Guanine Exchange Factor (GEF), the reverse process is facilitated by a GTPase-activation protein (GAP), which induces hydrolysis of the bound GTP [13]. D) MAPK (mitogen activated kinase) cascades consist of three or four different proteins (the kinases) that specifically catalyze the phosphorylation of the subsequent kinases under consumption of ATP. In this case, Raf is a MAPKKK, MEK is a MAPKK, and ERK is a MAPK. The number of phosphorylation events on each level can differ. Dephosphorylation is exerted by phosphatases (denoted by PP).

Figure 2

Signaling Pathways in Yeast. Overview of signaling pathways in the baker's yeast S. cerevisiae. From left to right: HOG pathway activated by osmotic shock, pheromone pathway activated by pheromones secreted by cells of the opposite mating type, pseudohyphal growth pathway stimulated by starvation conditions, and the glucose sensing pathway. In each pathway, the activated receptor activates a cascade of intracellular processes including complex formation, phosphorylation and dephosphorylation, and transport steps. Most of these pathways comprise a MAP kinase cascade and transcription factors that regulate the expression of target genes. Besides this, the signaling pathways can directly interact with cell cycle progression and adaptation of metabolism.

Figure 3

Jak-Stat Pathways. Jak-Stat pathways entail the receptor Janus kinase (Jak) and the signal transducer and activator of transcription (Stat). Stat molecules are inactive as monomers, and their activation involves phosphorylation and dimerization. The binding of the ligand (here the hormone Epo) to the receptor (EpoR) results in phosphorylation of Jak and of the cytoplasmatic domain of EpoR. Monomeric Stat5 is recruited to the phosphorylated and thereby activated receptor, EpoR_A. (1) Upon receptor recruitment, monomeric Stat5 is tyrosine-phosphorylated. It dimerizes (2) and migrates to the nucleus (3), where it binds to the promoter of target genes. After dissociation (4), it is dephosphorylated and exported to the cytoplasm.

Figure 4

WNT Pathways. The extracellular signaling molecule WNT activates three pathways: (1) Early cell fate decisions are controlled via the canonical pathway (middle): it comprises the regulation of gene expression by inducing β-catenin-mediated transcriptional activation. Interaction of WNT with the transmembrane receptor frizzled (FZ) activates dishevelled (DVL), which induces the disassembly of a complex consisting of axin, adenomatosis polysis coli (APC), glycogen synthase kinase 3β (GSK3β) and β-catenin. In non-stimulated cells, GSK3β phosphorylates β-catenin, thereby triggering its degradation. Activitation of the pathway effectively increases the levels of β-catenin in the cyctoplasm, which is then translocated to the nucleus. Here it it forms the β-catenin-T-cell specific transcription factor complex that activates the transcription of target genes. (2) In the planar cell polarity pathway (left), FZ functions through G-proteins to activate DVL, which thereupon signals to Rho GTPases (Rho or Rac or both). Activated Ras signals through the c-Jun amino (N)-terminal kinase (JNK). Activation of Rho-GTPases induces changes in the cytoskeleton. In neurons, this pathway is involved in dendritic arborization. (3) In the WNT/calcium pathway (right), activation of DVL activates protein kinase C (PKC) and induces the release of intracellular calcium, which activates a calcium/calmodulin-dependent protein kinase II (CaMKII).

Figure 5

Calcium-calmodulin network. Ca²⁺-signaling is mediated through several Ca²⁺-binding proteins, including calmodulin (CaM) and protein kinase C (PKC). The activity of N-methyl-d-aspartate (NMDA) receptors or voltage-sensitive Ca²⁺ channels leads to an increase in intracellular Ca²⁺, which triggers a release of calmodulin that was previously bound to neuromodulin or neurogranin. Depending on Ca²⁺, CaM modulates the activity of several key signaling molecules that are crucial for synaptic plasticity including adenylyl cyclases (AC), protein kinases, calcineurin, nitric oxide synthase, Ca²⁺-channels, ATP-dependent Ca²⁺-pumps, and the CaM-dependent protein kinases (CaMKII). CaM has four Ca²⁺ binding sites and in the presence of CaM binding protein, it shows heterotropic positive cooperativity for Ca²⁺. This enables that CaM-regulated AC and cyclic nucleotide phosphodiesterases have different Ca²⁺-sensitivities, and that CaM-stimulated phosphatase calcineurin has greater sensitivity to Ca²⁺ than CaMKII.

Figure 6

Signaling network in neurons Schematic overview of signaling pathway modules in neuronal cells (redrawn from Bhalla [82]). Abbreviations: RTK: receptor tyrosine kinase; mGluR: metabotropic glutamate recptor; GPCR: G-protein coupled receptor; NMDAR: N-methyl D-aspartate Receptor; Gq: G-protein type q; Gsα: G-protein type s; PLCβ: phospholipase C β; PLCγ: phospholipase C γ; IP3: inositol trisphosphate; DAG: diacylglycerol; Sos/GEF: Son of Sevenless/guanine nucleotide exchange factor; Ca²⁺ : Calcium; PKC: protein kinase C; AC: adenylyl cyclase; PDE: phosphodiesterase; CaM: calmodulin; CaMKII: calcium calmodulin kinase type II; cAMP: cyclic adenosine monophosphate; CaN: calcineurin; AA: arachidonic acid; PLA₂: phospholipase A2; MAPK: mitogen activated protein kinase; MKP-1: MAP-Kinase phosphatase type 1; PKA: protein kinase A; PP1: protein phosphatase type 1.