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Haugh Model and Modification 1

December 12, 2017

Haugh Model and Modification

I. INTRODUCTION

Human growth hormone (hGH) is a peptide hormone that is required for several

important signaling pathways in the human body, including cellular growth and

metabolism. These pathways also promote aging, genomic instability, as well as

impacting sleep cycles (Guevara-Aguirre 2011). Consequently, an understanding of the

interaction between hGH and its receptor protein on the cell surface is vital.

hGH has two binding sites, site 1 and site 2, and both sites bind to the same receptor.

This signaling acts through receptor homodimerization. However, each site has a

different affinity for the receptor. Site 1 has a much higher affinity for the hGH receptor

than site 2. The sequence of reactions can then be stated as followed. hGH binds to a

receptor on the cell surface at site 1. The unbound site 2 of hGH in this complex then

binds to another receptor on the cell surface, forming a homodimer made of 1 hGH

protein and 2 receptors. This dimer then promotes downstream signaling pathways by

recruiting Janus kinase 2 (Jak2) tyrosine kinases to other receptors.

The pathway is more complex and involves many other cellular mechanisms,

involving internalizations of cell-surface proteins, recycling of internalized receptors, and

synthesis of receptors. The model developed by Jason M. Haugh was created to

incorporate these ligand and receptor trafficking mechanisms (Haugh 2004). He sought to

match dose response data reported in literature by modeling proliferation signal

depending on the amount of hGH introduced to the cell, as well as including the ability to

model and predict inhibition and different mutations in hGH. Rate constants used in this


model were determined from previous literature and by adjusting the parameters so as to

minimize the differences between the model and raw data. Initial conditions were set at

2000 receptor proteins per cell. Consequently, the model developed accurately

represented the raw data and predicted limitations.

II. MODEL

The Haugh model uses 4 ordinary differential equations to calculate dimer formation

and thus signaling. The ODEs used describe the hGH receptors on the cell surface (R),

the complexes bound to site 1 of hGH on the cell surface (C), dimers made of 1 hGH and

2 receptor proteins (D), and internalized receptor proteins (Ri) (Figure 1B). The ODEs

account for receptor and complex endocytosis, internalized receptor recycling, receptor

degradation, and binding at the cell surface (Figure 1A). Dimer degradation is assumed to

happen the instant a dimer is internalized, captured in the rate constant ke. Rate constants

used were of that in the Haugh paper.

The model was successfully used to describe the effect of changing the affinity of site

1 in hGH on the proliferation signal produced, calculated as per equation A5 in the

Haugh model (Figure 2). As expected, the general shape of the curve is a bell curve; the

proliferation signal increases as hGH concentration increases because dimer formation

increases, until so much hGH is added that instead of forming dimers, multiple hGH

proteins bind receptors at site 1, which has a higher affinity for the receptor. When the

affinity of site 1 is mutated to be higher (blue dashed line), the maximum proliferation

signal is about the same, but the range of hGH variant that can hold that maximum

decreases (Figure 2). This happens because as hGH increases, because the affinity at site


1 is higher the protein is more inclined to bind at site 1. When site 1 affinity is decreased

(red dashed line), the maximum proliferation signal decreases, but this maximum is

achieved at much higher concentrations of hGH (Figure 2). This is because with a lower

affinity, the protein is not pushed as hard to bind at site 1, so more hGH must be added to

force multiple hGH proteins to be bound only at site 1.

Similarly, the model was also able to predict the effect on proliferation signal by

changing the site 2 affinity for the receptor by changing the value of the rate constant kx2

(Figure 4). Increasing the affinity by a factor of 10 (dashed blue line) relative to wildtype

hGH resulted in a slightly increased maximum proliferation signal because this is the rate

constant leading to dimer formation, thus at steady state more dimer can be formed

(Figure 4). The higher affinity also allowed the maximum proliferation signal to be

achieved for a greater range of hGH variant concentration because it increased the

likelihood of hGH binding two receptors. Decreasing the affinity of site 2 for the receptor

by 10-fold (dashed red line) similarly decreases the maximum proliferation rate and the

range of hGH concentration this maximum can be held out for. However, both variants

and wildtype hGH reach their maximum proliferation signal at 1 nM hGH (variant),

indicating that changing the affinity of site 2 does not have an impact on the amount of

hGH required for maximum dimer formation.

The model was also able to predict the relationship between proliferation signal and

receptor downregulation, which was almost linear (Figure 3). This means that as the

dimers forming increases, there is a linear increase in proliferation signal. This

conclusion can help in interpreting other results as well, by drawing the relationship

between dimer formation and signaling.


Antagonistic effects of hGH variants were also successfully described by

implementation of the Haugh model (Figure 5). Antagonism by K172A/F176A (blue

line) showed that much higher levels of hGH variant needed to be added to decrease

proliferation signal, as expected because the affinity of K172A/F176A is about 650x

lower than the site 1 affinity of wildtype hGH. A defective site 2 of the hGH antagonist

showed a decease in proliferation signal at lower concentration of hGH variant, because

of the inability of the antagonist to dimerize (red and yellow lines). Furthermore, a higher

site 1 affinity of the antagonist was confirmed to decrease dimerization as lower

concentrations of hGH as seen in the difference between H21A/R64K/E17A/G120R and

G120R (red and yellow lines).

Using anti-hGH antibodies, Fab fragments were also modeled as antagonists because

of their inability to form dimers (Figure 7). Proliferation signal was calculated for

different concentrations of Fab 5 (red line) and Fab 13E1 (blue line), and Fab 5 decreases

proliferation signal (and thus dimerization) at lower concentrations of Fab than Fab 13E1.

Fab 5 antagonism decreases the kx2 of hGH binding rather than inhibiting, which proved

to be a more dramatic antagonistic effect because kx2 is the only rate contributing to dimer

formation.

Furthermore, the agonistic effect of anti-hGH antibodies as monoclonal antibodies, or

MAbs, on the proliferation signal was also demonstrated by the model (Figure 6). MAb 5

(purple line) cannot form dimers, so the proliferation signal was always 0. MAb 3D9

(yellow line) and 263 (red line) both reach their maximum proliferation signal at 10 nM

MAb, which indicates that the fact both have the same site 1 affinity is important to when

max proliferation occurs. However, the maximum proliferation signal with MAb 263 is


much higher than that of MAb 3D9, indicating that higher rate constants, like kf1 and kx2,

result in increased dimerization. MAb 13E1 (blue line) has the highest proliferation

signal but requires a higher MAb concentration, indicating that a higher KD needs a

higher MAb concentration for maximum dimerization and confirms that a higher kf1 and

kx2 result in increased dimer formation.

Dimer fraction versus time and hGH concentration was also calculated using the

Haugh model and gave expected results (Figure 6).

The model employed several key assumptions upon implementation. To begin with,

hGH was assumed to have been in excess, so that [L] ~ [L]0. This assumption is valid

because enough hGH (or hGH variant) is added to the system to be able to ignore ligand

depletion, and was confirmed to cause no change in proliferation signal by the simple

addition of a differential equation (Figure 9A). Additionally, the internalized pool of

receptors was ignored by setting the krec/kdeg fraction to 0; thus, krec = 0 and the value of

kdeg is irrelevant. This assumption serves to simplify the model but may not be as accurate

a representation of signaling as it could be. Furthermore, the model depends upon

receptor proteins and complexes being internalized at the same rate, as well as dimers

being degraded upon internalization at a different rate.

The model also makes several assumptions about the binding of hGH to the receptor

proteins. Haugh assumes that hGH first binds at site 1, which makes sense because this

site has a higher affinity. The model also depends upon site 1 dissociating to break up the

dimer, followed by the “fast” dissociation of hGH and receptor protein at site 2.


ODEs were written using standard mass action methods, indicating an assumption that

the solution is well-mixed (although a common assumption, this may not be the most

accurate representation).

Although several assumptions were made and many rate constants fit to the data, the

model well represented the data and gave insight into the mechanism.

III. CONCLUSION

Generally, the Haugh model fit the data provided. However, this good fit was brought

about by starting with rate constants determined by data, and then “adjusting” them to

give a better fit—essentially fitting the data, which takes away from the significance of

how good the fit is. Additionally, while the ODEs incorporated recycling and degradation

of internalized receptor, krec was set to 0, which neglected this entire portion of the model.

This means that the entire point of modeling receptor protein is mute and leaves a huge

hole in what this model could do. There is also room for improvement in the basic

assumption of the solution being well-mixed, where one could factor in different

compartments. Furthermore, the representation of the dissociation of the complex formed

after dissociation of the dimer, i.e. receptor bound to hGH at site 2, as “fast” leaves room

for modification. What exactly is “fast,” how can that truly be modeled, and what is the

mechanism behind it?

That being said, the assumptions made allowed for the model to be easily (more or

less) implemented to provide insight into the mechanism, including what rate constants

were important, how binding sites affected signaling, and antagonist agonist by anti-hGH

antibodies. Without the simplifications made by Haugh, this model would have been at


worst impossible and at best much more complex to implement and understand, and

could have less significance in determining the mechanisms at play.

IV. MODIFICATION

The Haugh model presents an opportunity for modification where it models the

irreversible dissociation of the complex formed upon dissociation of the dimer (receptor

bound to hGH at site 2) as “fast.” “Fast” is not a rate and is likely missing steps in

between, and no truly irreversible reaction exists in biology. This modification

incorporates reversible dissociation of the dimer to form a second complex (C2), where

receptor protein is bound to site 2 of hGH, which can also reversibly dissociate to

unbound receptor and ligand (Figure 10A). This model accounts not only for reversible

reactions and a more specific method of dimer formation and dissociation but also for the

binding of receptor protein to the lower-affinity site 2 of hGH first. This modification is

an important addition because it provides another mechanistic insight and possible

explanation; even if it does not prove to be as accurate as the Haugh model, it could be

adjusted and applied to other mechanisms. The ODEs used to implement the modification

can be found in Figure 10B. I hypothesize that this modification will as a general trend

increase the dimer formation and thus proliferation signaling.

The actual results of the implementation of the modified model proved to be more

complex than my simple hypothesis. The modification has no noticeable effect on the

dimer fraction formed at different time points and hGH concentrations (Figure 11).

Additionally, the relationship between receptor downregulation and proliferation signal


remains the same, but this makes sense as it only shows the relationship between dimer

and signaling (Figure 12).

However, once the changes in the affinities of the binding sites are modeled, we see

interesting changes with the modification. With site 1, with a higher affinity the

modification does not change the initial increase in proliferation signal at lower

concentrations of hGH variant or the maximum cell proliferation. However, the

modification does change the range of hGH concentration that this maximum can be held

for (Figure 13). At a higher affinity, the modification shortens it, but at a lower affinity

the modifications lengthens this range. Furthermore at a low site 1 affinity, the

modification increases both the range of hGH concentration at maximum proliferation

signal as well as increases the maximum proliferation signal.

Analysis of site 2 affinity reveals an analogous trend. The modification did not impact

the increase in proliferation signal as hGH concentration increases at low concentrations,

which makes sense because the modification includes site 2 affinity binding for dimer

formation, but that does not greatly impact formation since the affinity is lower than that

of site 1. However, at high kx2 values, the modification decreases the range of hGH

variant that achieve maximum proliferation signal, while at low values of kx2 this range is

increased. Perhaps at high values of kx2 although more dimer could potentially be formed,

more dimer could dissociate through the modified mechanism.

Although the results did not necessarily agree with my hypothesis, they brought about

interesting insight that could prove useful in future modeling.

dR/dt = vs + krec[Ri] – kt[R] – kf1[L][R] + kr1[C] – kx2[C][R] + k-x2[D] + 2k-x1[D]

dC/dt = kf1[L][R] - kr1[C] – kt[C] – kx2[C][R] + k-x2[D]

dD/dt = kx2[C][R] - k-x2[D] – k-x1[D] – ke[D]

dRi/dt = kt([R] + [C]) – krec[Ri] – kdeg[Ri]


V. FIGURES

Figure 1: The Haugh Model. Shown is the model developed and by Jason M. Haugh

and used to model hGH signaling data. Shown in red brackets is an illustration of a

reaction that is defined by Haugh as happening at a “fast” rate. Consequently, this

reaction is assumed to happen instantaneously once the dimer has unbound hGH at site 1.

(A) An illustration of the model is presented. (B) The ODEs used to implement the

mathematical model are shown.

hGH variant (nM)

Proliferation signal Haugh Model and Modification 10

Figure 2: Effect of site 1 affinity on proliferation signal. The Haugh model was

implemented to model the proliferation signal as a function of initial ligand (hGH

variant) concentration. Proliferation signal at steady state as a function of initial

concentration of wildtype hGH (solid black curve; site 1 KD = 1.5 nM), a high-affinity

hGH variant (dashed blue curve; H21A/R64K/E174A; site 1 KD = 0.05 nM), and a low

affinity hGH variant (dashed red curve; K172A/F176A; site 1 KD = 1.0 uM) were

modeled. The high affinity variant increases proliferation signal at a lower than wildtype

hGH, but this higher affinity also makes the proliferation signal remain high for a shorter

amount of time than wildtype. The lower affinity requires a much higher concentration of

hGH variant for maximum proliferation signal, and this maximum is lower than that of

wildtype hGH.

Receptor downregulation (%)

Proliferation signal


Figure 3: The relationship between proliferation signal and receptor

downregulation in wildtype hGH. Receptor downregulation is equal to initial receptor

proteins on the cell surface – the sum of the complex, dimer, and recepetor proteins on

the cell surface at steady state. This relationship between proliferation signal and

downregulation is almost linear and shows that maximum proliferation signal is achieved

at about 90% receptor downregulation.


Figure 4: Effect of site 2 affinity on proliferation signal. The proliferation signal at

steady state was calculated using the Haugh model for wildtype hGH (solid black line; kx2

= 0.0024, k-x2 = 0.016) and hypothetical hGH variants with changed site 2 affinities: one

with a 10x higher affinity (dashed blue line; kx2 = 0.024, k-x2 = 0.016) and one with a 10

times lower affinity (dashed red line; kx2 = 0.00024, k-x2 = 0.016). All 3 variants reach

peak proliferation signal around the same concentration of hGH, but the lower the affinity

(or the higher the kx2) at site 2 the higher the maximum proliferation signal. Furthermore,

lower affinity at site 2 also showed that a higher proliferation signal meant that more

hGH variant had to be added in order to decrease proliferation signal (due to self-

antagonism).


hGH variant (nM)


Figure 5: Antagonistic effect of hGH variants on proliferation signal. Proliferation

signal at steady state was calculated for cells with 1 nm hGH and different concentrations

of hGH variants. The variants used were K172A/F176A (blue line; site 1 KD = 1 uM),

wildtype hGH (purple line; site 1 KD = 1.5 nm), G120R (yellow line; site 1 KD = 1.5 nm,

kx2 = 0), and H21A/R64K/E17A/G120R (red line; site 1 KD = 0.5 nm, kx2 = 0). The higher

affinity site 1 with defective site 2 variant (H21A/R64K/E17A/G120R) requires less hGH

variant to be added than the normal affinity site 1 with defective site 2 variant (G120R).

Additionally, a defective site 2 shows less hGH variant required to decrease proliferation,

as shown by the difference between G120R and wildtype hGH. Contrarily, a normal site

2 but decreased affinity at site 1 requires more hGH variant to be added to decrease cell

proliferation, shown by the difference between wildtype hGH and K172A/F176A.

hGH variant (nM)



Antibody kf1 kr1 kx2 k-x2 k-x1

13E1 0.0024 0.042 0.0108 0.0042 0.0042

263 0.006 0.03 7.95x10^(-4) 0.003 0.003

3D9 0.002 0.01 2.04x10^(-5) 0.001 0.001

5 0.004 0.002 0 0.002 0.002

Figure 6: Effect of monoclonal antibodies as agonists on proliferation signal. (A)

Kinetic rates for the antibodies selected were calculated. (B) Proliferation signal was

calculated for four monoclonal antibodies, where dimers formed with the monoclonal

antibodies also contribute to the proliferation signal. The four selected anti-hGH

antibodies were 13E1, 263, 3D9, ad 5 (kinetic rates given in chart). MAb 5 cannot form

A.


Monoclonal antibody (nM)

B.


dimers (kx2 = 0), so the proliferation signal remained at 0 no matter the MAb

concentration. MAbs 3D9 and 263, both of which do not bind specifically at the hGH

binding site, reach peak proliferation signal around 10 nm MAb, but MAb 263 requires

much more monoclonal antibody added to decrease proliferation signal from its

maximum. Furthermore MAb 3D9 has a lower maximum proliferation signal.

Additionally, MAb 13E1 does not reach its peak maximum proliferation signal until 100

nM MAb, and this maximum is the highest of the four MAbs.


Figure 7: Effect of Fab fragments as antagonists on proliferation signal. Proliferation

signal at steady state was calculated at 1 nM hGH and varying concentrations of Fab

fragments. 2 Fab fragments were examined in particular: Fab 13E1 (red line; inhibition

with rate constants in chart with kf1/2 and kx2 = 0) and Fab 5 (blue line; changes the kx2 of

wildtype hGH). Fab 5 decreases proliferation signal at lower concentrations than Fab

13E1.


Fab concentration (nM)


Figure 8: Receptor dimer formation at different at different time points and hGH

concentrations. Dimers formed were calculated using the Haugh model at 1 minute, 10

minutes, 45 minutes, and steady state. At each time, hGH concentration was 0.01 (dark

blue), 0.1 (cerulean blue), 1 (light blue), 10 (teal), 100 (light green), 103 (orange), and 104

(yellow) nM. At 0.01 nM hGH, the dimer fraction does not increase significantly until 10

minutes, and the increases slightly at 45 minutes until it reaches its steady state value. At

0.1 nM, dimer fraction increases 5x between 1 and 10 minutes, and remains at a dimer

fraction of 0.1 between 10 minutes and 45 minutes, but then eventually decreases about

50% to its steady state value. At 1 nM, the dimer fraction peaks at 10 minutes at a value


of 0.45, but then decreases to 0.075 at 45 minutes, and then again to about 0.03 at steady

state. At 10 nM hGH, the dimer fraction peaks at 1 minute at a value of 0.6, but decreases

slightly to 0.4 at 10 minutes, then decreases to about 0.3 for steady state. At 100 nM

hGH, the dimer fraction remains relatively constant for the first 10 minutes at 0.3, then

drops to 0.75 at 45 minutes, and then again to about 0.025 at steady state. At 103 nM

hGH, the dimer fraction increases slightly after one to ten minutes to 0.6 but hovers

around its steady state value. At 104 nM hGH, there is not much of a difference between

the dimer fraction at steady state and 1 minute. Steady state dimer fraction is at a

maximum at 1 and 10 nM hGH.


Figure 9: Ignoring ligand depletion is a valid assumption in the Haugh Model. To

check the ligand depletion assumption, an adjustment of the Haugh model was made and

hGH variant (nM)


A.

B.


compared to the original. (A) Proliferation signal at steady state was calculated at varying

concentrations of hGH using the original Haugh model (black line) and with an adjusted

model that factors for ligand depletion. Proliferation signal factoring in ligand depletion

plotted against the initial hGH concentration (blue circles) and plotted against the actual

concentration (subtracting the hGH depleted; red crosses) were also calculated. This

adjustment was no different than the original Haugh model, demonstrating this was a

valid assumption. The actual difference in cell proliferation was on the order of 10-30. (B)

The adjusted ODEs to factor in ligand depletion are shown. The added ODE is for hGH

(L), where Nav = Avogadro’s number and Ce = cells/mL, assumed to be 4 x 1012.


Figure 10: Modification of the Haugh Model. Shown is a modification of model

developed and by Jason M. Haugh. (A) The modification is shown in red brackets.

Instead of a “fast” and irreversible dissociation of hGH bound to receptor at site 2, this

modification accounts for a reversible reaction with a rate constant associated with it.

This modification also allows for complex and dimer formation for when hGH binds to

the receptor at site 1 first. (B) The ODEs used to implement the modified model are

shown. This modification assumes the same values for rate constants as in the Haugh

paper and also assumes that affinity of site 1 for the receptor remains unchanged whether

or not site 2 is bound, and vice versa. Thus, it is assumed that k-x3 = kr1, kx3 = kf1, k-x4 = k-

x2, and kx4 = kx2.

dR/dt = vs + krec[Ri] – kt[R] – kf1[L][R] + kr1[C] – kx2[C][R] + k-x2[D] + k-x3[D] – kx3[R][C2] + k-x4[C2] – kx4[R][L]

dC/dt = kf1[L][R] - kr1[C] – kt[C] – kx2[C][R] + k-x2[D]

dD/dt = kx2[C][R] - k-x2[D] – k-x3[D] + kx3[R][C2] – ke[D]

dRi/dt = kt([R] + [C] +[C2]) – krec[Ri] – kdeg[Ri]

dC2/dt = -kt[C2] + kx4[R][L] – k-x4[C2] – kx3[R][C2] + k-x3[D]


A.

B.


Figure 11: Modified Haugh model does not noticeably change dimers fraction at

different times and hGH concentraions. Dimers formed were calculated 1 minute, 10

minutes, 45 minutes, and steady state. At each time, hGH concentration was 0.01 (dark

blue), 0.1 (cerulean blue), 1 (light blue), 10 (teal), 100 (light green), 103 (orange), and 104

(yellow) nM. (A) As shown in Figure 8, the original Haugh model was used to calculate

the dimer fractions. (B) The modified Haugh model was used to calculate dimer

fractions, which appear the same as those of the original Haugh model.


Figure 12: Modification of Haugh model does not change the relationship between

proliferation signal and receptor downregulation. Receptor downregulation was

calculated as per equations provided by Jason M. Haugh where receptor downregulation

equals initial receptor proteins on the cell surface – the sum of the complex, dimer, and

recepetor proteins on the cell surface at steady state. The proliferation signal versus

downregulation calculated by the original Haugh model (light blue line) perfectly

overlaps with that of the modified model (dashed red line).Proliferation signal

Receptor downregulation (%)


Figure 13: Comparison of the effect of site 1 affinity on proliferation signal in the

modified and original Haugh model. Proliferation signal at steady state was calculated

for different concentrations of hGH variant using the original Haugh or modified model,

as labeled. In black is the wildtype hGH (site 1 KD = 1.5 nM), where the solid line

represents the original model and the dashed line represents the modified model. In blue

is the hGH variant H21A/R64K/E174A with a high affinity site 1 (KD = 0.05 nM), where

the solid line represents the original model and the dashed line represents the

modification. In red is the hGH variant K172A/F176A with a low affinity site 1 (KD = 1.0

uM), where the solid line represents the original model and the dashed line represents the

modification.

hGH variant (nM)



Figure 14: Comparison of the effect of site 2 affinity on proliferation signal in the

modified and original Haugh model. The proliferation signal at steady state was

calculated for wildtype hGH (black lines; kx2 = 0.0024, k-x2 = 0.016) and hypothetical

hGH variants with changed site 2 affinities: one with a 10x higher affinity (blue lines; kx2

= 0.024, k-x2 = 0.016) and one with a 10 times lower affinity (red lines; kx2 = 0.00024, k-x2

= 0.016). Solid lines represent calculations done using the original Haugh model, and

dashed lines represent calculations done using the modified Haugh model.

hGH variant (nM)



References

Jason M. Haugh. “Mathematical Model of Human Growth Hormone (hGH)-Stimulated

Cell Proliferation Explains the Efficacy of hGH Variants as Receptor Agonists or

Antagonists.” Biotechnology Progress 20 (2004): 1337-1344.

Jaime Guevara-Aguirre, Priya Balasubramanian, Marco Guevara-Aguirre, Min Wei,

Federica Madia, Chia-Wei Cheng, David Hwang, Alejandro Martin-Montalvo, Jannette

Saavedra, Sue Ingles, Rafael de Cabo, Pinchas Cohen, Valter D. Longo. “Growth

Hormone Receptor Deficiency Is Associated with a Major Reduction in Pro-Aging

Signaling, Cancer, and Diabetes in Humans.” Science Translational Medicine 3 (2011):

1-9.

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