Abstract
Cytoplasm contains a large number of macromolecules at extremely high densities. How this striking nature of intracellular milieu called macromolecular crowding affects intracellular signaling remains uncharacterized. Here, we examined the effect of macromolecular crowding on ERK MAPK phosphorylation by MEK MAPKK. Addition of polyethylene glycol6000 (PEG6000) as a crowder to mimic intracellular environments, elicited a biphasic response to the overall ERK phosphorylation rate. Furthermore, probability of processive phosphorylation (processivity) of tyrosine and threonine residues within the activation loop on ERK increased nonlinearly for increasing PEG6000 concentration. Based on the experimental data, we developed for the first time a mathematical model integrating all of the effects of thermodynamic activity, viscosity, and processivity in crowded media, and found that ERK phosphorylation is transitionstatelimited reaction. The mathematical model allows accurate estimation of the effects of macromolecular crowding on a wide range of reaction kinetics, from transitionstatelimited to diffusionlimited reactions.
Introduction
Cells sense a variety of extracellular signals by receptors, and input signals to a system of intracellular reaction network, i.e., an intracellular signal transduction cascade. These input signals are processed by the intracellular signal transduction cascades to drive the cells to exhibit specific phenotypes. A collapse of the system by gene mutations leads to pathological outcomes, including autoimmune disease and cancer^{1}. Therefore, a better understanding of the intracellular signal transduction cascade is one of the critical issues for the control of diseases. The intracellular signal transduction cascade consists of a chain of diffusion, proteinprotein interaction and enzymatic reactions, and thus it can be described by ordinary and partial differential equations, and solved numerically in order to analyze its dynamics^{2,3,4}. Such a systems biology approach with experimentallyverified kinetic parameters provides a basis for understanding the signal transduction cascade^{5,6,7,8}.
The cytoplasm is highly crowded with macromolecules, such as proteins, lipids, nucleic acids and so on^{9,10}. The concentration of proteins in the cytoplasm is over 100 mg/ml, with macromolecules making up 30–40% of the cytoplasmic volume^{11,12}. This congested condition, often referred to as macromolecular crowding, results in enzymatic reaction kinetics that differ significantly from those described previously in idealdilute solution. For example, macromolecular crowding affects the enzymatic reaction rate through two factors: thermodynamic activity and viscosity^{13,14,15,16}. The former factor, thermodynamic activity, compensates for the concentration in an ideal solution to yield an effective concentration in a real solution^{15,17}. Macromolecular crowding increases thermodynamic activity mainly through the excluded volume effect, and consequently leads to an increase in a reaction rate. The latter factor, viscosity, reduces an encounter rate between the enzyme and the substrate^{15,18}. Therefore, the reaction rate is negatively correlated with viscosity.
Processive reactions can be observed in phosphorylation^{19,20}, polyubiquitination^{21}, and so on. These multisite posttranslational modifications of signaling proteins are a common mechanism by which a complex cellular phenotype can be exhibited^{22}. We demonstrated by a computer simulation that crowding environment increased the processivity in twostep phosphorylation reactions of ERK MAP kinase^{23}. This prediction was validated by our experimental results^{24}.
Although macromolecular crowding has been suggested to have a great influence on the intracellular signaling pathway, it is not fully understood how the macromolecular crowding affects the overall reaction rate and processivity of enzymes. Here, we propose mathematical models of reaction kinetics that account for the effect of thermodynamic activity, viscosity and processivity in an environment with macromolecular crowding, and provide empirical validations by employing in vitro ERK MAP kinase phosphorylation.
Results
Quasiprocessive phosphorylation of ERK under macromolecular crowding
We have recently shown that Tyr and Thr residues within the activation loop of ERK MAP kinase were phosphorylated by MEK in a quasiprocessive manner in mammalian cells^{24}. In the first step, MEK binds to and phosphorylates ERK to generate tyrosine monophosphorylated ERK (pYERK), and then dissociates from pYERK (Fig. 1A). Under macromolecular crowding, diffusion of MEK and pYERK is restricted. Therefore, rebinding of the MEK and the pYERK, which will not substantially happen in ideal solutes, occurs with some probability, and consequently the MEK processively phosphorylates pYERK, producing the tyrosine and threonine bisphosphorylated ERK (pTpYERK).
We analyzed the effects of macromolecular crowding by simplified quasiprocessive phosphorylation model as follows. First, the rate constants of two successive phosphorylation reactions of ERK in the absence of a crowder agent were defined as k_{1} and k_{2}, respectively (Fig. 1B, left). Our previous analysis demonstrated that these rate constants were firstorder parameters, since the Michaelis constants in these reactions were significantly higher than the concentration of substrate, npERK, used in this study^{24}. Under this condition, the k_{1} and k_{2} values can be determined by fitting experimental data (Fig. 1B, left). Second, we built a simplified quasiprocessive model, taking the crowding factor (c) and processivity (p) into consideration; c was defined as a dimensionless value that compensates for the increase in or decrease of reaction rates under macromolecular crowding in comparison with the reaction rates in the idealdilute solution. c reflects the effects of both thermodynamic activity and viscosity on the reaction rate^{18}. Therefore, the rate constants in all reactions should be multiplied by c under a crowding condition (Fig. 1B, right). p is defined as the probability of a processive reaction; i.e., the probability that pTpYERK will be generated directly from npERK (Fig. 1B, right). The second reaction of processive phosphorylation would be on the order of 10–0.1 msec^{23}, and much faster than k_{1} and k_{2}. For this reason, the reaction rate of the processive reaction can be described as k_{1}*p. According to this definition, the rate constant of pYERK generation must be multiplied by (1p). With the k_{1} and k_{2} values determined in the absence of a crowder, c and p values can be obtained by fitting the experimental data with the quasiprocessive model^{24}.
Measurements of crowding factor and processivity by in vitro ERK phosphorylation in the absence or presence of a crowder, polyethylene glycol 6000
To evaluate the effect of macromolecular crowding on c and p, the kinetics of ERK phosphorylation in vitro were studied in the absence or presence of a crowder agent, polyethylene glycol 6000 (PEG6000)^{24}. A constitutively active mutant of MEK1 (MEK) and a kinasedead mutant of GSTfused ERK2 (GSTERK) were used as the kinase and substrate of the in vitro phosphorylation reaction, respectively. Samples were subjected to phosphoaffinity gel electrophoresis with Phostag compound, which clearly separates four phosphoisoforms of ERK (Fig. 2A). The fractions of phosphoisoforms of ERK, namely, nonphosphorylated ERK (npERK), pYERK and pTpYERK, were quantified by an Odyssey Infrared Imaging System (Fig. 2B, and Supplementary Fig. S1 online). Under our experimental condition, the order of electrophoretic mobility in the Phostag gel was, from the slowest to fastest, pYERK, pTpYERK, pTERK (if any), and npERK^{25,26}. Two phosphorylation rates, k_{1} and k_{2}, were determined from the data obtained in the absence of PEG6000 (Fig. 2B, left). c and p at each concentration of PEG6000 were determined by fitting the experimental data with the simplified quasiprocessive model (Fig. 1B, right and Supplementary Fig. S1 and Dataset online).
PEG6000 showed a dosedependent biphasic effect on c with the highest acceleration at approximately 10% (v/v) (Fig. 2C, green dots). This result is in agreement with the scheme documented by Minton et al.^{18}. p increased steeply with increasing PEG concentration (Fig. 2C, red dots).
Formulation of the crowding effect on the overall reaction rate
We first formulated the effect of crowding on the overall reaction rates, namely, c in this study. Minton and coworkers have established a basis for the mathematical framework of macromolecular crowding. Here, a biphasic effect on the enzymatic reaction rate is explained by increasing excluded volume and viscosity^{18}. To formulate them, a simple enzymatic reaction is considered: where E, S, P and ES represent the enzyme, substrate, product and enzymesubstrate transition state complex.
When the encounter rate between E and S is much higher than the conversion rate of ES to E + P, the reaction is said to be transitionstatelimited. Under such a condition, the overall rate constant, k_{f}, is obtained by where k_{ts} is the transitionstatelimited rate constant, and k^{0}_{ts} is the limiting value of k_{ts} in the absence of an added crowder^{13,27}. Γ is defined by where γ_{E}, γ_{S}, and γ_{ES} are the activity coefficients of the enzyme, substrate and enzymesubstrate transition state complex, respectively^{27}. The crowder and enzyme are considered as hard spherical particles with radius r_{C}, and r_{E}. The activity coefficient of the enzyme in a fluid of the crowder occupying fraction φ of total fluid volume is given by where and Similarly, the activity coefficients of the substrate and enzymesubstrate complex, γ_{S} and γ_{ES}, are also obtained with their radii, r_{S} and r_{ES}. We calculated the Γ value in the presence of crowder agents of different sizes, 5 kD, 20 kD, and 100 kD (Fig. 3A). For this calculation, we assumed that all molecules were rigid spherical particles, in which radii were proportional to the cube root of their molecular weights. Consistent with previous reports^{27}, the Γ value increased monotonously with increasing volume fraction of crowder agent and correlated inversely with the molecular weight of the crowder^{17}.
When the encounter rate between the enzyme and substrate is small relative to the conversion rate of enzymesubstrate complex to product, the reaction is said to be diffusionlimited. Under this condition, the encounter rate, k_{enc}, which is proportional to the sum of the diffusion coefficients of enzyme and substrate, dictates the rate of reaction. The diffusion coefficient is inversely proportional to the viscosity of the solvent according to the StokesEinstein relationship. As an empirical approximation^{28,29}, we may describe the rate of reaction with an exponential decay of crowder concentration where k^{0}_{enc} is the encounter rate constant in the absence of a volumeexcluding background species, and g is a constant value that is a function of the relative sizes and shapes of the enzyme, substrate and crowder molecule^{27}. We quantified the relative viscosity of PEG6000 to water by an Ostwald viscometer, and obtained a g value of 0.11 by fitting with the exponential function (Fig. 3B). Based on Eq. 6, the overall rate constant decreased exponentially with an increasing fraction of PEG volume under a diffusionlimited condition.
Eqs. 1 and 6 did not recapitulate the biphasic effect of the overall reaction rate independently of each other. To the best of our knowledge, the biphasic effects as observed in the experimental results have not been explicitly described by a mathematical framework, though such opposing effects of thermodynamic activity and viscosity on reaction rate have been well documented^{15,18,28}. We considered an intermediate condition between a transitionstatelimited reaction and a diffusionlimited reaction. Noyes^{30} indicates the relationship between the conversion and encounter rates as follows: Combining Eqs. 1, 6 and 7 gives the following equation: Using the overall reaction rate k^{0}_{f} and ratio of the encounter rate to conversion rate θ, Eq. 8 is written as where and Based on the definition of c, we derived an equation of c as follows: Importantly, under the transitionstatelimited reaction (θ≫ 1), Eq. 9 corresponds to Eq. 1, whereas under the diffusionlimited reaction (θ≪ 1) Eq. 9 gives Eq. 6. According to the value of Γ and the g value in PEG6000 solution, we calculated the c value at different concentrations of the crowder with a changing θ value as a variable (Fig. 3C, green lines). These results provided reasonable fits to the effect of crowder on the overall reaction rate in both the transitionstatelimited and diffusionlimited reaction. By fitting the experimental data shown in Fig. 2C to this model, we obtained a θ value of 2.26 (Fig. 3C, bold green line). The value implicated the transitionstatelimited reaction to some extent in ERK phosphorylation by MEK.
Formulation of processivity
We have defined the processivity as a probability of twostep successive phosphorylation reactions from npERK to pYERK to pTpYERK by an identical MEK. To formulate the processivity, we consider the situation where ERK has just been phosphorylated by MEK and converted to pYERK. Particles of MEK and pYERK are treated as pointlike particles. These particles diffuse normally at all times in this model. Since MEK contains only a single ATPbinding pocket^{31}, after the phosphorylation reaction, MEK must release ADP and bind to ATP in a manner dependent on an ordered bibi reaction. We define the period required for the replacement of ADP with ATP as a relaxation time, τ_{rel}. The threedimensional probability distribution of pYERK after τ_{rel} is then given by where r is the distance from MEK and D is the sum of the diffusion coefficient of MEK and pYERK in crowder solvent (Fig. 4A). We assume a reaction radius, r_{reac}, in which MEK rebinds to and processively phosphorylates pYERK within a time of τ_{rel}. The processivity, p, is then given by where Erf is an error function given by Taylor series of an error function and exponential function are given by
Based on Eqs. 17 and 18 for small x ≪ 1, the Eq. 15 becomes The relations in Eq. 19 were consistent with a part of the results reported previously^{23}; the rebinding dynamics of MEK and pYERK is derived from a threedimensional random walker returning to the origins. Because we simplified MEK and pYERK as point sources, we could not obtain the t^{−1/2} decay dynamics of processivity in a range of times shorter than the time to travel a molecular diameter^{23}. The diffusion coefficient is inversely proportional to the viscosity of the solvent according to the StokesEinstein relationship. Thus, Eq. 15 becomes where D_{water} is the sum of the diffusion coefficient of MEK and pYERK in water. To evaluate this model, we calculated processivity according to Eq. 20. D_{water} was set at 10 μm^{2}/sec based on the viscosity of water and the size of MEK and ERK. In this model, we did not consider the effect of viscosity on diffusion coefficient of ADP/ATP, i.e. τ_{rel}, because of their high concentration and fast diffusion (see Discussion). We set the reaction radius r_{reac} at 10 nm, which was comparable to the sum of the radius of MEK and ERK molecules. Processivity was obtained by changing the value of τ_{rel}, namely, the reactivation time of MEK, and plotted against the concentration of PEG6000, which increased viscosity (Fig. 4B). The processivity elevated with increasing concentration of PEG6000. Furthermore, the model demonstrated a best fit to the experimental data at the τ_{rel} value of 0.020 msec (Fig. 4B, bold line).
A full reaction kinetics model of ERK phosphorylation
In the quasiprocessive phosphorylation model described in this study, the association and dissociation of ERK to MEK were neglected for the simplicity. Therefore, we performed similar analyses based on the full reaction kinetics of ERK phosphorylation model. Kinetic parameters were fitted with experimental data in the absence of PEG6000 (Fig. S2A, left), followed by the fitting of crowding factor, c, and dissociation constant, k_{b−}, in the full model with experimental data under the condition of PEG6000 (Fig. S2A right). Of note, the reaction model can strictly define processivity as k_{2}/(k_{b−} + k_{2}) without any need to introduce an empirical parameter of processivity in the simplified model (Fig. 1B). The full model recapitulated time course of ERK phosphorylation in the absence or presence of PEG6000 (Fig. S2B). The fitted values of crowding factor (Fig. S2C) and processivity (Fig. S2D) provided consistent values of θ (1.80) and τ_{rel} (0.024 msec) in Eq. 12 and Eq. 20, respectively.
Analysis of crowding factor and processivity based on the scaled particle theory
The crowding effect (Eq. 12) and processivity (Eq. 20) explicitly include viscosity, which is macroscopic parameter described by the StokesEinstein law. Therefore, we next examined the microscopic effect of crowder on crowding factor and processivity by estimating diffusion coefficients based on the scaled particle theory (SPT)^{32}. Let r_{t} represent the radius of the tracer species, Δr′ the approximated length of the Brownian displacement^{32}. The scaled particle theory yields where D and D_{water} are the diffusion coefficient of tracer species in the presence or absence of background crowder. A_{i} is given by where r_{b} is radius of background molecule, M is molecular weight, N_{a} is Avogadro's number, c_{back} is a concentration of background molecules [g/litter], and ν is in cm^{−3}. We obtained almost same result of BSA selfdiffusion as previously reported in Muramatsu and Minton, indicating the valid calculation (Fig. S3A). Next, logarithmic ratio of D to D_{water} of relative diffusion coefficient of MEK and ERK with the increase in PEG6000 concentration was estimated by SPT (Fig. S3B). The microscopic model with SPT significantly underestimated the relative diffusion coefficient in the presence of PEG6000 (Fig. S3B, red line) in comparison to the relative diffusion coefficients calculated by the experimental data with Ostwald viscometer (Fig. S3B, blue line). We recalculated and fitted the crowding factor (Fig. S3C) and processivity (Fig. S3D) using SPT with the experimental data; however there existed significant discrepancy between the experimental data and mathematical model with the scaled particle theory. This could be due to steeper reduction of the relative diffusion coefficient between MEK and ERK obtained by SPT. From these results, we concluded, at this time, that the increase in viscosity under the condition of macromolecular crowding was the responsible factor for the crowding factor and processivity. The clear dissection of difference between microscopic and macroscopic effect of molecular crowding is one of challenging tasks for future studies. Simulation of the enzymatic reaction at the particle level will enable to directly address this issue^{23}.
Analytical solution of the simplified quasiprocessive phosphorylation model
Finally, based on Eq. 12 and 20, analytical solutions of npERK, pYERK and pTpYERK concentration, [npERK], [pYERK], and [pTpYERK], respectively, in the quasiprocessive model (Fig. 1B) are given by where the [npERK], [pYERK], and [pTpYERK] at time = 0 are ERK_{0}, 0, and 0, respectively. k_{1} and k_{2} were firstorder successive phosphorylation rates [/sec] of ERK by MEK based on our previous study^{24}. To simplify these equations, we omitted the factor of MEK concentration, and included it implicitly in k_{1} and k_{2}. In general, analytical solutions of two successive reactions were obtained at c = 1 and p = 0 as follows: In the quasiprocessive phosphorylation model, we assumed that the reaction rate of processive phosphorylation is equivalent to k_{1}*p, because the second reaction of processive phosphorylation would be much faster than k_{1} and k_{2}. These equations provided a good agreement with the dynamics of ERK phosphorylation at different concentrations of PEG6000 (Fig. 5), supporting the validity of the model.
Discussion
In this study, we provided for the first time the mathematical framework for the rate constant and processivity at intermediate conditions between transitionstatelimited and diffusionlimited reactions under the condition of molecular crowding. Moreover, we derived an analytical expression for the kinetics in ERK processive phosphorylation, demonstrating time courses of phosphorylation as a function of crowder concentration, φ. These models recapitulated reaction kinetics of dual phosphorylation of ERK by MEK with or without crowder agent. However, slight, but not ignorable, disagreement is also evident between the experimental data and the model, especially in late phase of higher concentrations of PEG6000, 12% and 18%. This result suggests that the PEG6000 solution demonstrates a spatially heterogeneous environment, which entails the introduction of a timedependent rate coefficient with a fractal kinetics model^{13,33}. In a previous simulation study, diffusion obstacles gave rise to anomalous diffusion^{34}. Consistent with this idea, anomalous diffusion in the cytoplasm was also observed by fluorescence correlation spectroscopy^{35,36,37}. Moreover, it has recently been indicated that fractal structures exist in the nucleus^{38,39}. These findings should pave the way for future work that will improve the model more quantitatively.
We demonstrated a clear dependency of processivity in Eq. 19 on diffusion coefficient, D, reactivation time, τ_{rel}, and reactivation radius, r_{reac}. Membraneanchoring or transmembrane proteins are wellknown to show slower diffusion coefficients in comparison to those of cytoplasmic proteins^{40}. Of note, many of receptor tyrosine kinases (RTKs) such as epidermal growth factor receptor form homo and heterodimer upon ligand stimulation on membrane, and transphosphorylates multiple sites of tyrosine residues in RTKs^{41}, strongly suggesting the possibility of processive phosphorylation in the RTK activation. Diffusioncontrolled ligandreceptor kinetics supports this possibility^{42}. Furthermore, rebinding event would take place in protein translocation on nucleic acid^{43,44}. Intriguingly, by analogy to the diffusion coefficient, clustering or confinement of molecules in small compartment also induced the increase in processivity^{45}.
Taking the high concentration and fast diffusion rate of intracellular ATP molecules into consideration, the ratelimiting step of ADP/ATP exchange, i.e. reactivation time, τ_{rel}, is likely to be determined by the ADP release from MEK. The dissociation rate constants of ADP from kinases have been reported to vary over a wide range with an order of magnitude of 3 to 4 (0.23 sec^{−1} to 140 sec^{−1})^{46,47,48,49,50}. The ADP/ATP exchange rate predicted in this study, 50000 sec^{−1}, which is the reciprocal of τ_{rel} is much larger than the dissociation rate constants reported previously. There are at least two possible explanations for this discrepancy. First, the reaction radius, r_{reac}, may have been underestimated in this study. Currently, we do not have any methods to quantitatively estimate the reaction radius. Second, the phosphorylation reaction model may not be correct. We employed an ordered bibi sequential reaction of processive phosphorylation, in which MEK dissociated from pYERK, followed by exchange of ADP with ATP to phosphorylate pYERK. If MEK processively phosphorylates ERK through a random bibi mechanism, the processivity will increase even under a slow ADP/ATP exchange rate. In fact, Horiuchi et al. have indicated that the mechanism of MEK turnover involves random addition of substrate and ATP^{51}. Although they have also suggested the ordered release of product from MEK^{51}, effects of crowding on the mechanism of MEK turnover have not been fully characterized, and should be considered in a future study.
The mathematical models developed here integrate three effects of molecular crowding on the reaction rate—namely, thermodynamic activity, viscosity and processivity. These models were consistent with experimental data supporting the processive ERK phosphorylation by MEK. Our findings will contribute to the effort to quantitatively simulate intracellular signal transduction.
Methods
Plasmids
pGEX4T3ERK2KR, a kinasedeficient mutant (K57R) of Xenopus ERK2, and pET12HisFLAGMEK1SDSE, a constitutively active mutant (S218D/S222E) of Xenopus MEK1, have been purified as described previously^{24}.
Reagents and antibodies
Phostag acrylamide was obtained from the Phostag Consortium (http://www.phostag.com/). Fifty percent polyethylene glycol 6000 (PEG6000) solution was purchased from Hampton Research (Aliso Viejo, CA). AntiERK1/2 and antiphosphoERK1/2 (Thr202/Tyr204 and Thr185/Tyr187, respectively) were purchased from Cell Signaling Technology (Beverly, MA). LICORblocking buffer and IRDye680 and IRDye800conjugated antirabbit and antimouse IgG secondary antibodies were obtained from LICOR Bioscience (Lincoln, NE).
Purification of recombinant proteins and in vitro kinase assay
GSTERK2KR and 12HisMEK1SDSE were prepared as described previously^{24}. For the in vitro kinase assay, 1 μM GSTERK2, 0.1 μM 12HisFLAGMEK1SDSE and different concentrations of PEG6000 (0~20% v/v) were incubated in in vitro kinase buffer (50 mM Tris [pH 7.5], 10 mM MgCl_{2}, 0.02% BSA 0.2 mM DTT). The reaction was started by adding 5 × ATP solution (5 mM ATP, 50 mM MgCl_{2}, 25 mM Tris [pH 7.2], 0.15 M NaCl) and stopped at the indicated time point by adding SDS sample buffer.
Phostag polyacrylamide gel electrophoresis
Phostag polyacrylamide gel electrophoresis was performed essentially as described previously^{24,25}. Briefly, 50 μM Phostag and 100 μM MnCl_{2} were added to conventional SDSpolyacrylamide separation gels according to the manufacturer's protocol, respectively. Proteins were detected by using an Odyssey Infrared Imaging System (LICOR).
Measurements of viscosity
Viscosities of the aqueous PEG6000 solutions relative to water were determined by using an Ostwald viscometer at 37°C in a water bath.
Parameter search
Numerical simulation and parameter search were implemented by MATLAB software with ode15s function and fminsearch function, respectively.
References
 1.
Hanahan, D. & Weinberg Robert, A. Hallmarks of Cancer: The Next Generation. Cell 144, 646–674 (2011).
 2.
Kholodenko, B. N. Cellsignalling dynamics in time and space. Nat. Rev. Mol. Cell Biol. 7, 165–176 (2006).
 3.
Kitano, H., Funahashi, A., Matsuoka, Y. & Oda, K. Using process diagrams for the graphical representation of biological networks. Nat. Biotechnol. 23, 961–966 (2005).
 4.
Schoeberl, B., EichlerJonsson, C., Gilles, E. D. & Muller, G. Computational modeling of the dynamics of the MAP kinase cascade activated by surface and internalized EGF receptors. Nat. Biotechnol. 20, 370–375 (2002).
 5.
Fujioka, A. et al. Dynamics of the Ras/ERK MAPK cascade as monitored by fluorescent probes. J. Biol. Chem. 281, 8917–8926 (2006).
 6.
MatsunagaUdagawa, R. et al. The scaffold protein Shoc2/SUR8 accelerates the interaction of Ras and Raf. J Biol. Chem. 285, 7818–7826 (2010).
 7.
Kamioka, Y., Yasuda, S., Fujita, Y., Aoki, K. & Matsuda, M. Multiple decisive phosphorylation sites for the negative feedback regulation of SOS1 via ERK. J Biol. Chem. 285, 33540–33548 (2010).
 8.
Aoki, K., Nakamura, T., Inoue, T., Meyer, T. & Matsuda, M. An essential role for the SHIP2dependent negative feedback loop in neuritogenesis of nerve growth factorstimulated PC12 cells. J. Cell Biol. 177, 817–827 (2007).
 9.
Minton, A. P. How can biochemical reactions within cells differ from those in test tubes? J Cell Sci. 119, 2863–2869 (2006).
 10.
Medalia, O. et al. Macromolecular architecture in eukaryotic cells visualized by cryoelectron tomography. Science 298, 1209–1213 (2002).
 11.
Fulton, A. B. How crowded is the cytoplasm? Cell 30, 345–347 (1982).
 12.
Zimmerman, S. B. & Trach, S. O. Estimation of macromolecule concentrations and excluded volume effects for the cytoplasm of Escherichia coli. J. Mol. Biol. 222, 599–620 (1991).
 13.
Schnell, S. & Turner, T. E. Reaction kinetics in intracellular environments with macromolecular crowding: simulations and rate laws. Prog. Biophys. Mol. Biol. 85, 235–260 (2004).
 14.
AlHabori, M. Macromolecular crowding and its role as intracellular signalling of cell volume regulation. Int. J. Biochem. Cell Biol. 33, 844–864 (2001).
 15.
Ellis, R. J. Macromolecular crowding: an important but neglected aspect of the intracellular environment. Curr. Opin. Struct. Biol. 11, 114–119 (2001).
 16.
Ellis, R. J. Macromolecular crowding: obvious but underappreciated. Trends Biochem. Sci. 26, 597–604 (2001).
 17.
Hall, D. & Minton, A. P. Macromolecular crowding: qualitative and semiquantitative successes, quantitative challenges. Biochim. Biophys. Acta 1649, 127–139 (2003).
 18.
Zimmerman, S. B. & Minton, A. P. Macromolecular crowding: biochemical, biophysical, and physiological consequences. Annu. Rev. Biophys. Biomol. Struct. 22, 27–65 (1993).
 19.
Koivomagi, M. et al. Cascades of multisite phosphorylation control Sic1 destruction at the onset of S phase. Nature 480, 128–131 (2011).
 20.
Pellicena, P. & Miller, W. T. Processive phosphorylation of p130Cas by Src depends on SH3polyproline interactions. J. Biol. Chem. 276, 28190–28196 (2001).
 21.
Williamson, A. et al. Regulation of ubiquitin chain initiation to control the timing of substrate degradation. Mol. Cell 42, 744–757 (2011).
 22.
Patwardhan, P. & Miller, W. T. Processive phosphorylation: mechanism and biological importance. Cell Signal. 19, 2218–2226 (2007).
 23.
Takahashi, K., TanaseNicola, S. & ten Wolde, P. R. Spatiotemporal correlations can drastically change the response of a MAPK pathway. Proc. Natl. Acad. Sci. U. S. A. 107, 2473–2478 (2010).
 24.
Aoki, K., Yamada, M., Kunida, K., Yasuda, S. & Matsuda, M. Processive phosphorylation of ERK MAP kinase in mammalian cells. Proc. Natl. Acad. Sci. U. S. A. 108, 12675–12680 (2011).
 25.
Kinoshita, E., KinoshitaKikuta, E., Takiyama, K. & Koike, T. Phosphatebinding tag, a new tool to visualize phosphorylated proteins. Mol. Cell Proteomics. 5, 749–757 (2006).
 26.
Kinoshita, E., KinoshitaKikuta, E. & Koike, T. Phostag SDSPAGE systems for phosphorylation profiling of proteins with a wide range of molecular masses under neutral pH conditions. Proteomics 12, 192–202 (2012).
 27.
Minton, A. P. Molecular crowding: analysis of effects of high concentrations of inert cosolutes on biochemical equilibria and rates in terms of volume exclusion. Methods Enzymol. 295, 127–149 (1998).
 28.
Minton, A. P. Excluded Volume as a Determinant of Macromolecular Structure and Reactivity. Biopolymers 20, 2093–2120 (1981).
 29.
Caldin, E. F. & Hasinoff, B. B. Diffusioncontrolled kinetics in the reaction of ferroprotoporphyrin IX with carbon monoxide. Journal of the Chemical Society, Faraday Transactions 1: Physical Chemistry in Condensed Phases 71, 515–527 (1975).
 30.
Noyes, R. M. Effects of Diffusion Rates on Chemical Kinetics. Progress Reaction Kinetics. Peter G., ed. an editor. Pergamon Press, New York. 1, 129–160 (1961).
 31.
Fischmann, T. O. et al. Crystal structures of MEK1 binary and ternary complexes with nucleotides and inhibitors. Biochemistry (Mosc.) 48, 2661–2674 (2009).
 32.
Muramatsu, N. & Minton, A. P. Tracer diffusion of globular proteins in concentrated protein solutions. Proc. Natl. Acad. Sci. U. S. A. 85, 2984–2988 (1988).
 33.
Kopelman, R., Parus, S. & Prasad, J. Fractallike exciton kinetics in porous glasses, organic membranes, and filter papers. Phys. Rev. Lett. 56, 1742–1745 (1986).
 34.
Saxton, M. J. Anomalous diffusion due to obstacles: a Monte Carlo study. Biophys. J. 66, 394–401 (1994).
 35.
Wachsmuth, M., Waldeck, W. & Langowski, J. Anomalous diffusion of fluorescent probes inside living cell nuclei investigated by spatiallyresolved fluorescence correlation spectroscopy. J. Mol. Biol. 298, 677–689 (2000).
 36.
Malchus, N. & Weiss, M. Elucidating anomalous protein diffusion in living cells with fluorescence correlation spectroscopyfacts and pitfalls. J. Fluoresc. 20, 19–26 (2010).
 37.
Weiss, M., Hashimoto, H. & Nilsson, T. Anomalous protein diffusion in living cells as seen by fluorescence correlation spectroscopy. Biophys. J. 84, 4043–4052 (2003).
 38.
Bancaud, A. et al. Molecular crowding affects diffusion and binding of nuclear proteins in heterochromatin and reveals the fractal organization of chromatin. EMBO J. 28, 3785–3798 (2009).
 39.
LiebermanAiden, E. et al. Comprehensive mapping of longrange interactions reveals folding principles of the human genome. Science 326, 289–293 (2009).
 40.
Reits, E. A. & Neefjes, J. J. From fixed to FRAP: measuring protein mobility and activity in living cells. Nat. Cell Biol. 3, E145–147 (2001).
 41.
Jones, R. B., Gordus, A., Krall, J. A. & MacBeath, G. A quantitative protein interaction network for the ErbB receptors using protein microarrays. Nature 439, 168–174 (2006).
 42.
Goldstein, B. & Dembo, M. Approximating the effects of diffusion on reversible reactions at the cell surface: ligandreceptor kinetics. Biophys. J. 68, 1222–1230 (1995).
 43.
Zhou, H. X. A model for the mediation of processivity of DNAtargeting proteins by nonspecific binding: dependence on DNA length and presence of obstacles. Biophys. J. 88, 1608–1615 (2005).
 44.
Berg, O. G. & von Hippel, P. H. Diffusioncontrolled macromolecular interactions. Annu. Rev. Biophys. Biophys. Chem. 14, 131–160 (1985).
 45.
Mugler, A., Bailey, A. G., Takahashi, K. & ten Wolde, P. R. Membrane clustering and the role of rebinding in biochemical signaling. Biophys. J. 102, 1069–1078 (2012).
 46.
Jan, A. Y., Johnson, E. F., Diamonti, A. J., Carraway, I. K. & Anderson, K. S. Insights into the HER2 receptor tyrosine kinase mechanism and substrate specificity using a transient kinetic analysis. Biochemistry (Mosc.) 39, 9786–9803 (2000).
 47.
Shaffer, J., Sun, G. & Adams, J. A. Nucleotide release and associated conformational changes regulate function in the COOHterminal Src kinase, Csk. Biochemistry (Mosc.) 40, 11149–11155 (2001).
 48.
Callaway, K., Waas, W. F., Rainey, M. A., Ren, P. & Dalby, K. N. Phosphorylation of the transcription factor Ets1 by ERK2: rapid dissociation of ADP and phosphoEts1. Biochemistry (Mosc.) 49, 3619–3630 (2010).
 49.
Shaffer, J. & Adams, J. A. Detection of conformational changes along the kinetic pathway of protein kinase A using a catalytic trapping technique. Biochemistry (Mosc.) 38, 12072–12079 (1999).
 50.
Lew, J., Taylor, S. S. & Adams, J. A. Identification of a partially ratedetermining step in the catalytic mechanism of cAMPdependent protein kinase: a transient kinetic study using stoppedflow fluorescence spectroscopy. Biochemistry (Mosc.) 36, 6717–6724 (1997).
 51.
Horiuchi, K. Y., Scherle, P. A., Trzaskos, J. M. & Copeland, R. A. Competitive inhibition of MAP kinase activation by a peptide representing the alpha C helix of ERK. Biochemistry (Mosc.) 37, 8879–8885 (1998).
Acknowledgements
We thank the members of the Matsuda Laboratory for their helpful discussions. KA and MM were supported by Research Program of Innovative Cell Biology by Innovative Technology (Cell Innovation) from the Ministry of Education, Culture, Sports, and Science, Japan. KA was supported by the JST PRESTO program and JSPS KAKENHI (21790273).
Author information
Affiliations
Laboratory of Bioimaging and Cell Signaling, Graduate School of Biostudies, Kyoto University, Sakyoku, Kyoto 6068501, Japan
 Kazuhiro Aoki
 & Michiyuki Matsuda
PRESTO, Japan Science and Technology Agency (JST), 418 Honcho Kawaguchi, Saitama 3320012, Japan
 Kazuhiro Aoki
Laboratory for Biochemical Simulation, RIKEN Quantitative Biology Center, 623 Furuedai, Suita, Osaka, 5650874, Japan
 Koichi Takahashi
 & Kazunari Kaizu
Institute for Advanced Biosciences, Keio University, 5322 Endo, Fujisawa, 2520882, Japan
 Koichi Takahashi
Department of Pathology and Biology of Diseases, Graduate School of Medicine, Kyoto University, Sakyoku, Kyoto 6068501, Japan
 Michiyuki Matsuda
Authors
Search for Kazuhiro Aoki in:
Search for Koichi Takahashi in:
Search for Kazunari Kaizu in:
Search for Michiyuki Matsuda in:
Contributions
K.A., K.T., K.K. and M.M. designed research; K.A. performed experiments; K.A., K.T. and K.K. analyzed data; K.A., K.T., K.K. and M.M. wrote the paper.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to Kazuhiro Aoki.
Supplementary information
PDF files
 1.
Supplementary Information
Supplementary Information
Excel files
 1.
Supplementary Information
Supplementary Dataset
Rights and permissions
This work is licensed under a Creative Commons AttributionNonCommercialNoDerivs 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/byncnd/3.0/
To obtain permission to reuse content from this article visit RightsLink.
About this article
Further reading

Singlecell quantification of the concentrations and dissociation constants of endogenous proteins
Journal of Biological Chemistry (2019)

Kinetic and mechanistic studies of p38α MAP kinase phosphorylation by MKK 6
The FEBS Journal (2019)

Factors defining the effects of macromolecular crowding on dynamics and thermodynamic stability of heme proteins invitro
Archives of Biochemistry and Biophysics (2018)

Stress granule formation via ATP depletiontriggered phase separation
New Journal of Physics (2018)

Crowding, Entropic Forces, and Confinement: Crucial Factors for Structures and Functions in the Cell Nucleus
Biochemistry (Moscow) (2018)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.