# A Computational Study of the Effects of Syk Activity on B Cell Receptor Signaling Dynamics

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## Abstract

**:**

## 1. Introduction

## 2. Model Development

#### 2.1. Biological Background

#### 2.2. Model

#### 2.3. Model Equations

**BCR Activation**: The receptor dynamics considered here include engagement of the BCR, ITAM tyrosine phosphorylation, Syk binding to the BCR, and BCR internalization, recycling and degradation. The key variables include free BCR x

_{BCRfree}, BCR bound by ligand x

_{BCRb}, singly-phosphorylated BCR x

_{BCRp}

_{1}, and doubly-phosphorylated BCR x

_{BCRp}

_{2}. The model formulation reflects how the kinase Syk can bind to either form of the phosphorylated BCR. Due to the positive promotion of ITAM tyrosine phosphorylation by membrane proximal PTKs [11], we assume that if Syk binds to a singly-phosphorylated BCR, that receptor will become doubly-phosphorylated before the kinase can unbind. Thus there is no term for unbinding from Sykb to BCRp1 in any of the model equations. Receptor internalization x

_{BCRi}is promoted by clathrin, which is localized ${x}_{Clathri{n}_{local}}$ to the membrane by Syk.

**Syk Activation:**We consider four forms of Syk, three of which have been modified through binding or phosphorylation. The variable x

_{Syk}represents the amount of kinase that has not been activated and is unbound. The variable x

_{Sykb}is the basally active form of the kinase that has been bound to the BCR. The catalytically active form of Syk that has been phosphorylated at tyrosine Y342 and Y346 is denoted by x

_{Syk}

_{342}and is responsible for enhancing signaling propagation. If either active form of the kinase becomes phosphorylated at tyrosine Y317 it is rendered inactive. This inactive form is denoted by x

_{Syk}

_{317}. The forms represented by x

_{Syk}

_{342}and x

_{Syk}

_{317}are still assumed to be bound to the BCR. As discussed below, each of these four forms of Syk can bind to an orthogonal inhibitor; this binding also renders Syk inactive.

**Syk-AQL dynamics:**As discussed in the Introduction, Syk-AQL allows for Syk activity to be modulated through the addition of the orthogonal inhibitor (OI). The binding of the OI to the mutant Syk-AQL is also modeled using mass action kinetics and can be seen in Figure 2.

**Lyn Activation:**For the Src-family PTK Lyn (x

_{Lyn}) to become fully activated (x

_{Lyn}

_{*}), it must be dephosphorylated at Y508 (x

_{Lyndp}) and then go through an autophosphorylation reaction. We consider both events with the following equations:

**Regulatory Enzyme Dynamics:**Following the initiation of BCR signaling, the regulation of BCR, Syk, and Lyn activity is orchestrated by feedback loops involving the aforementioned PTKs and a collection of regulatory enzymes. The dynamic members of the regulatory subsystem are SHP1, Csk and Cbp, with their dynamics being driven by the amount of CD45. The activated forms of SHP1, Csk and Cbp are denoted by the variables x

_{SHP}

_{1*}, x

_{Csk}

_{*}and x

_{Cbpp}

_{*}, respectively, and are modeled with the following equations:

**Medial Signaling Dynamics (BLNK, BTK, PLC2γ):**The second messenger PLC2γ is critical for transducing a signal downstream following Syk activation. Before becoming fully activated, PLC2γ must bind to the linker protein BLNK and be phosphorylated by Syk and the Bruton’s tyrosine kinase (BTK). Here BTK must also bind to BLNK, and it is phosphorylated by Syk and Lyn before it becomes fully activated. These events are modeled by the following equations:

## 3. Materials and Methods

#### 3.1. Experimental Protocols

#### 3.1.1. Cell Lines

#### 3.1.2. Cellular Activation Assay

#### 3.2. Sensitivity Analysis

_{k}. Note that p

_{k}is the k

^{th}point in our parameter screening. To estimate the uncertainty in the data σ

_{out}for Equation (1), we assumed a linear dependence of σ

_{out}on y

_{obs}and conducted a linear regression using the information in Figure 2C of [8]. We found that the error in the measurements could be reasonably approximated by

_{obs}is an observed measurement. Given the total number of model parameters and the cost associated with varying them, we partitioned parameters into seven distinct groups and conducted a sensitivity analysis with respect to each group when determining which parameters to screen initially. These groups of parameters were detrained using natural divisions such as BCR dynamics, Syk dynamics, regulatory enzyme and Lyn dynamics, Erk pathway dynamics, etc. A study by Zheng [2] comparing local derivative-based sensitivity methods and global variance-based methods found that global parameter sensitivities were necessary to capture model behavior when considering a large parameter space, but that there were no significant difference between Sobol analysis and the other variance based methods considered. Given the relative independence of these groups, we calculated only primary Sobol sensitivities [17] to estimate the sensitivity of the outputs, normalized as in Equation (2), to each specified parameter. The sensitivity, ${S}_{{p}_{k}}(t)=\frac{Va{r}_{{p}_{k}}({E}_{p\ne {p}_{k}}[{x}_{out}^{stimulation}|{p}_{k}])}{Var({x}_{out}^{stimulation})},$ for a given parameter was computed at integer values t = 0, …, 30 using the method based on sparse-grid interpolation as described in [18]. This expression is designed to capture the relative sensitivity of the output as a function of one particular parameter p

_{k}, averaged over the other parameters. That is, if we fix p

_{k}, we can determine the average behavior as we vary the remaining parameters, and then determine how this average changes as a function of p

_{k}. These calculations were carried out in log space in each parameter, with a range of one order of magnitude above and below the nominal value for each parameter.

#### 3.3. Parameter Screening

_{out}| ≤ η or equivalently

_{sim}in that we calculate phosphorylation relative to the ending value rather than the basal value. That is, the signal intensities for the Western blots from our data were normalized by their ending phosphorylation levels to avoid the magnification of errors that would result from a small initial value. Applying the same normalization procedure to simulated data gives the form

#### 3.4. Contour Analysis

## 4. Results

#### 4.1. Sensitive Parameters

_{kf}specifically represents the forward rate of the ligand binding reaction to the BCR. Group two contains parameters related to Syk activation, and rw7

_{kr}is the reverse rate of the phosphorylation reaction for the Y342 tyrosine on Syk. Parameter rw9

_{kf}is the forward rate in the phosphorylation reaction for the Y317 tyrosine on Syk that has already been phosphorylated at Y342. Group three is comprised of parameters from the regulatory enzyme subsystem. A sensitivity analysis was not conducted with respect to this group due to issues with stiffness. We describe next steps to examine this stiffness and future plans to expand the regulatory enzyme subsystem to become fully dynamic in Section 5.

_{kf}is the rate at which BLNK is phosphorylated by Syk342, while rw16

_{kf}and rw16

_{kr}are the forward and backward rates for the binding of PLCγ to the linker protein BLNK. In group five, r12s

_{kf}is the forward rate at which Syk342 phosphorylates bound PLCγ. Parameters r13

_{kf}and r13

_{kr}represent the rate at which PLCγ phosphorylates the phospholipid PIP

_{2}and the corresponding rate of dephosphorylation. Finally, group six is made up of medial signaling parameters for reactions involving the kinase PKC and also the downstream MAPK pathway leading to Erk. Parameter r18

_{kf}is the rate at which Erk is phosphorylated by MEK. Parameter r19

_{kf}is the rate at which the enzyme SOS binds to phosphorylated BLNK. Finally, Group seven consists of parameters for reactions related to the NFκB pathways. Here r38

_{kf}is the rate of phosphorylation of IκB by the kinase IKK.

#### 4.2. Parameter Screening and Fitting

_{Erkp}| ≤ 1 and |J

_{IκB}

_{(}

_{dose}

_{1)}| ≤ 2. We found seven parameter vectors that met the criteria among the 1800 candidates considered. Due to large per-simulation time requirements, large objective values for doses #3 and #4 of the dose response experiments, and tradeoffs between Erk costs and IκB costs, we determined we would need to manually tune parameters related to IκB to achieve reasonable fits at all four doses.

_{kf}allowed us to fit two of the three nonzero data points. The agreement to the mutant data with this new parameter vector ${p}_{Mutant}^{*}$ can be seen in the right panel of Figure 5. Intuitively, this corresponds to inhibiting a larger fraction of Syk, and thus there is less Syk available to propagate a signal. Interestingly, we could also achieve the same fits to mutant data by lowering the total amount of Syk in the cell. This was reminiscent of the effects of the drug tetracycline, which can regulate the amount of kinase prior to stimulation. Note that the measurements used from Oh et al. were reported relative to phosphorylation levels observed following an experiment where cells were stimulated using PMA and ionomyocin. We do not simulate the effects of ionomyocin in this work since calcium is not modeled, so our simulated activity in the right panel of Figure 5 is relative to the final phosphorylation level observed in simulated wild-type activity.

#### 4.3. Contour Analysis

_{kr}> −0.5 is reasonably described as a function of log rw0

_{kf}− α log rw0

_{kr}. This leads to a kind of power law affinity, ${K}_{a,\alpha}=\frac{rw{0}_{kf}}{{(rw{0}_{kr})}^{\alpha}}$, where the multiplier α = 3/4 is the reciprocal of the slope of the linear relationship in the contour plot. The origin of the power law affinity will be investigated in future analysis of the dynamical system.

_{kf}and the responses in the high reverse rate region against the power law affinity. As expected from the contour plots, the plots in Figure 8 show a clear dependence on forward rate alone in the region of low reverse rate and a reasonably clear dependence on the power law affinity in the region of high reverse rate.

_{Sykb}, x

_{Syk}

_{342}, and x

_{317}all follow a sigmoidal course. Note that a percentage of total Syk is also allocated to other variables, such as free, unbound Syk, and to Syk bound to clathrin; since our focus is on the active forms of Syk, we omit these other forms. We find that Syk-AQL with no orthogonal inhibitor mimics fairly closely the response of wild-type, except that Syk342 is somewhat reduced. As expected, Syk-AQL with orthogonal inhibitor shows a marked decrease in these three forms of Syk, with the balance migrating to inhibited Syk.

#### 4.4. Independent Dataset Comparison

## 5. Conclusions and Future Directions

_{kr}our responses actually depended on quantities other than the standard affinity constant. For low levels of rw0

_{kr}, the model predicts that the response depends only on the forward rate of BCR binding rw0

_{kf}. At higher levels of rw0

_{kr}, the model predicts that the response depends on a power law form of the affinity constant, ${K}_{a,\alpha}=\frac{rw{0}_{kf}}{{(rw{0}_{kr})}^{\alpha}}$. These predictions were robust for WT and mutant simulations. Given the complexity of the dynamical system, a model reduction will likely be necessary in order to analytically investigate the origin of the power law affinity underlying the model response.

^{2+}dynamics, and the addition of the NFAT pathway. We plan also to restructure the dynamics of CD45, which is constant in the current version of the model; this modification will impact the regulatory enzyme dynamics as they are driven by CD45 activity.

## Supporting Information

processes-03-00075-s001.pdf## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**A depiction of the early, medial, and downstream signaling events induced by binding between B cell receptor and ligand, as described in the biological background section. Jagged arrows denote stimulations, curved arrows denote binding, straight arrows denote conversions, and color denotes species to appear repeatedly in the diagram. The plus and minus marks near the IκB-NF-κB disassociation reaction indicate which are positive feedbacks and which are negative feedbacks.

**Figure 3.**Simulations using ${p}_{WT}^{*}$ compared with experimental data. On the left, a simulation for the Erkp time course (normalized by total Erk) with 20 µg/mL anti-BCR is shown with the mean from Healy et al. [8] at time t = 5 and one standard deviation interval of uncertainty. On the right, simulations using ${p}_{WT}^{*}$ and 10 µg/mL anti-BCR (normalized by Erk at time t = 15) are compared with Erkp triplicate data from Section 3.1.2.

**Figure 4.**Simulations using ${p}_{WT}^{*}$ compared with IκB data from Healy et al. [8]. Simulations for non-degraded IκB (normalized by total IκB) are shown (left to right, top to bottom) for 5.5, 16.5, 50 and 150 µg/mL anti-BCR, with all measurements taken at time t = 15 and one standard deviation interval of uncertainty.

**Figure 5.**Anti-BCR dose response curves compared with experimental data from Oh et al. [1]. On the left, a simulation using ${p}_{WT}^{*}$ (normalized by WT activity at the maximum dose) is shown to qualitatively agree with the wild-type NF-κB data (•). On the right, a simulation with the parameter vector ${p}_{Mutant}^{*}$ (also normalized by WT activity at the maximum dose) is shown with NF-κB data (•) from B cells with Syk-AQL activity.

**Figure 6.**Anti-BCR dose response curves for baseline Syk-AQL activity and inhibited activities. The curves show simulated relative activity for Erkp measured at t = 5 after applying ligand and orthogonal inhibitor (µM) simultaneously. All curves have been normalized by Erk activity at the maximum dose with no orthogonal inhibitor added. The color of the curve corresponds to the amount of orthogonal inhibitor specified in the legend.

**Figure 7.**Contour plots for wild-type (WT), mutant without orthogonal inhibitor and mutant with 1 µM orthogonal inhibitor. The diagonal black line has a slope equal to 4/3. Regions with high values correspond to large Erkp response and small NF-κB response (both responses normalized by their maximum WT activity), and hence possible regions of anergy. Both rates are shown in log scale.

**Figure 8.**Plots of normalized Erkp minus normalized NF-κB (each normalized by their maximum WT activity) over a product grid of forward and reverse binding rates as in the contour plots above, but separated into regions of high and low reverse rates. The first column is wild-type simulation, the second column is mutant simulation without orthogonal inhibitor, and the third column is mutant simulation with 1 µM orthogonal inhibitor. Rates are shown in log scale.

**Figure 9.**Plots for three forms of Syk in the model as a function of the power law affinity constant for wild-type and mutant behavior. We notice lower phosphorylation levels at both tyrosine Y317 and Y342 in the mutant cells. After the addition of 1 µM orthogonal inhibitor to the mutant cell there is the expected decrease in overall activity; the balance is accounted for by inactive forms of Syk.

**Figure 10.**Anti-BCR dose response curves resulting from ${p}_{WT}^{*}$; the figure shows ligand dose response for Erkp resulting from ${p}_{WT}^{*}$ as compared with data from Chaudhri et al. [6]. As with the data, the simulation curve is normalized by the simulated value at the maximum dose 0.5 µg/mL.

Reactions | Parameters | |
---|---|---|

Group 1 | BCR dynamics | rw0_{kf} |

Group 2 | Syk activation | rw7_{kr}, rw9_{kf} |

Group 3 | Regulatory enzyme dynamics | N/A |

Group 4 | Medial signaling | rw15_{kf}, rw16_{kf}, rw16_{kr} |

Group 5 | Medial signaling | r12s_{kf}, r13_{kf}, r13_{kr} |

Group 6 | Erk pathway dynamics | r18_{kf}, r19_{kf} |

Group 7 | NF-κB pathway dynamics | r38_{kf} |

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**MDPI and ACS Style**

McGee, R.L.; Krisenko, M.O.; Geahlen, R.L.; Rundell, A.E.; Buzzard, G.T.
A Computational Study of the Effects of Syk Activity on B Cell Receptor Signaling Dynamics. *Processes* **2015**, *3*, 75-97.
https://doi.org/10.3390/pr3010075

**AMA Style**

McGee RL, Krisenko MO, Geahlen RL, Rundell AE, Buzzard GT.
A Computational Study of the Effects of Syk Activity on B Cell Receptor Signaling Dynamics. *Processes*. 2015; 3(1):75-97.
https://doi.org/10.3390/pr3010075

**Chicago/Turabian Style**

McGee, Reginald L., Mariya O. Krisenko, Robert L. Geahlen, Ann E. Rundell, and Gregery T. Buzzard.
2015. "A Computational Study of the Effects of Syk Activity on B Cell Receptor Signaling Dynamics" *Processes* 3, no. 1: 75-97.
https://doi.org/10.3390/pr3010075