- The document models the adsorption of fibrinogen proteins onto a hydrophobic surface. It considers the two orientations - side-on and end-on - that fibrinogen can bind in. - The model describes the adsorption as a series of reactions using ordinary differential equations. It takes into account the initial binding, unbinding, and rearrangement between the two orientations. - The model is nondimensionalized to reduce the number of parameters and make it dimensionless. This allows the adsorption profiles from the model to be compared to experimental data on fibrinogen adsorption.

1. Modelling the Adsorption of Fibrinogen and its Changes
of Orientation Due to Surface Chemistry
Elizabeth Mott
September 2016
Dissertation Submitted in Partial Fulfilment of the degree of MSc in programme title (year 1)
I warrant that the content of this dissertation is the direct result of my own work and that
any use made in it of published or unpublished material is fully and correctly referenced.
Abstract
The influence of a protein’s shape and chemistry, with respect to its adsorption onto
a surface, is presented. The case of the protein molecule, fibrinogen, binding to a hy-
drophobic surface was modelled, as the shape of a fibrinogen molecule is an oblong
shape rather than a spherical. The adsorption of a fibrinogen molecule binding to a
hydrophobic surface occurs in two stages, because of its oblong shape. Fibrinogen has
two orientations in which it can bind to a surface upon adsorption and rearrangements
of orientation can occur after the initial adsorption stage. The rate of initial adsorption
and orientation rearrangement has been defined by modified tanh functions dependent
on the surface’s chemistry. It was found that the model gives rise to adsorption profiles
similar to those determined experimentally.
1 Introduction
Adsorption is the accumulation of molecules or particles onto a surface [7]. The adsorption
of protein molecules from blood to a bio-material’s surface is an important study in material,
and medical science, where a bio-material is a synthetic material (usually a plastic) suitable
for implanting in a living body to repair damaged or diseased parts e.g. hip replacements
[11].
The process of adsorption is a complicated one which depends upon a variety of factors. The
size of the protein affects adsorption [4]. Smaller proteins can diffuse through solution more
quickly; however, they have fewer contact points with the surface, so the strength of the ad-
sorption is weaker [9]. The chemistry of the protein molecules, and the surface, has an influ-
ence on the rate and strength of adsorption. Both the protein and the surface can be either
hydrophobic or hydrophilic. A molecule is hydrophobic when it is repelled by water in a so-
lution and a molecule is hydrophilic when it is attracted to water in a solution. Hydropho-
bic protein molecules tend to bind more rapidly and form stronger bonds with a surface that
is hydrophobic, or charged. The strong attraction between a hydrophobic surface and a hy-
drophobic protein is driven by the protein molecules being repelled by water to the surface.
Both the protein and the surface is partly covered, and no longer in contact with as much
water in the solution [5], due to the adsorption of the protein onto the surface. On the other
1

2. Figure 1: A diagram representing the different orientations by which fibrinogen can bind to a
surface. Side-on orientation, S1 and end-on orientation, S2.
hand, hydrophilic proteins are not repelled by water in a solution so are less attracted to hy-
drophobic surfaces. It is important to note that despite the nature of hydrophobic and hy-
drophilic proteins stated above, a protein of either nature will still bind to surfaces with a
hydrophobic or hydrophilic charge, just at different rates and binding strengths.
The adsorption of blood proteins to biomaterial surfaces is an area of active research [2],
where fibrinogen and albumin are two known protein molecules studied in the blood. Albu-
min is a globular, spherical, protein that helps regulate both the binding of blood with phar-
maceuticals and the osmotic pressure of the blood [8]. Fibrinogen is a large oblong shaped
protein. Fibrinogen’s main role in the blood is to aid the formation of blood clots. Albumin
and fibrinogen can adsorb onto both hydrophobic and hydrophilic surfaces; however, both are
found to interact more strongly with hydrophobic surfaces [8].
The adsorption of fibrinogen onto a hydrophobic surface is complicated, compared to albu-
min. Fibrinogen binding appears to occur in steps, where the initial rate of adsorption is
rapid (50 seconds), whilst the second stage occurs over a longer period (60minutes) [8]. Fib-
rinogen has two binding orientations due to its oblong shape. When a fibrinogen protein ad-
sorbs onto a surface it can either bind with its long-axis perpendicular to the surface, which
is denoted the end-on orientation, or it can bind with its long-axis parallel to the surface, this
is denoted the side-on orientation. These orientations are shown in Figure 1. It has been sug-
gested that fibrinogen initially adsorbs in the side-on orientation quickly, covering all the sur-
face, then, after time, switches to the end-on orientation, uncovering binding sites and allow-
ing additional fibrinogen molecules to bind [8]. This rearrangement is said to be due to the
hydrophobic nature of fibrinogen, as the end-on orientations are attracted together so they
align with each other for minimal surface exposure to the water in the solution[8]. This two
staged adsorption of fibrinogen has only been observed at high concentrations and onto hy-
drophobic surfaces [8].
Protein-surface interactions have been previously modelled mathematically by [1], [3], [6] and
[10] using the law of mass action, yielding governing ordinary differential equations (ODEs),
which were then solved numerically.
The rates of two proteins adsorbing onto a surface have been compared by two different mod-
els in [6], where the first model was based on a ‘single lumped kinetic parameter’ and the
second model considered ‘the individual transport processes occurring prior to the adsorp-
tion reaction’. In the case when the protein molecule albumin was modelled, it was concluded
that neither approach correctly predicted the adsorption profiles observed from experimental
results. The poor fit using the kinetic rate constant model is suspected to be due to the com-
petition between the albumin molecules and the lysozyme molecules in the solution, as the
2

3. Figure 2: The adsorption profile and the derivative plot (inset) of fibrinogen on hydrophobic
(circle plot) and hydrophilic (triangle plot) terminated surfaces, reproduced, with permission
from; [8]. The two stages of adsorption can be seen more distinctively in the derivative, com-
pared to the adsorption profile.
albumin molecules are much larger than the lysozymes, hindering the adsorption of albumin.
In [3], the adsorption process of a protein (FNIII7−10) was modelled with respect to its con-
formational changes, where the protein’s bonds break down after adsorption, causing the pro-
tein to spread out across the surface. The conformational changes of the protein (FNIII7−10)
upon adsorption here was represented by parameter values for each of the molecule’s states,
a1 and a2, where a1 denotes the area occupied by the molecules before a conformational change
has occurred, state 1, and a2 is the area occupied after the conformational changes have oc-
curred, state 2. However, it was concluded that the conformational changes, by the use of
the parameter values a1 and a2, did not truly reflect the conformational change of the pro-
tein (FNIII7−10) when compared with experimental data. This is because the conformation
changes of a protein are governed by more than just the surface area occupied by a protein.
A study was also conducted in which two models were developed to investigate the adsorp-
tion characteristics of proteins with different concentrations, surface affinities, sizes and areas
[10]. It was concluded that the nature of proteins can be modelled using only information on
the protein’s concentration in a solution, its surface affinity and size.
The adsorption of fibrinogen molecules onto a surface can be modelled using a similar ap-
proach to [1], [3], [6] and [10]. The adsorption process of fibrinogen has also been investigated
experimentally [8], providing information that can aid in model formulation.
The information available on the adsorption of fibrinogen influenced the decision to model,
and replicate, the experimental results published in [8], showing the adsorption profile of fib-
rinogen onto a hydrophobic terminated surface. Since the shape of a molecule and the chem-
istry between a molecule and a surface is of great interest to those studying the adsorption
of molecules onto a surface, a hydrophobic surface is chosen to be modelled as the chem-
istry between fibrinogen molecules and a hydrophobic surface is what causes the change of
orientation and produces a two staged adsorption profile for fibrinogen, as seen in Figure 2.
Therefore, the models formulated in the remainder of this paper focus on the case of a single
protein species, fibrinogen and take into account the different orientations of fibrinogen.
3

4. 2 Model 1 formulation: constant reaction rates
Fibrinogen molecules adsorb onto a surface, binding to sites on the surface in either the side-
on, S1, orientation or the end-on, S2, orientation, where the side-on orientation will take up
more surface area than those in the end-on orientation. Once the molecules have adsorbed
onto the surface, they may flip between orientations. When a protein flips from the S1 to the
S2 orientation it opens up free binding sites, S. Taking all this into account, the following
reactions must be modelled, where C is the amount of protein in the solution:
C + S
k+
1
k−
1
S1 , (1)
C + S
k+
2
k−
2
S2 , (2)
S1
k+
3
k−
3
S2 + S. (3)
The binding rate of fibrinogen in the side-on (end-on) orientation is denoted as k+
1 (k+
2 ),
whilst the rate of the unbinding is denoted as k−
1 (k−
2 ). The rate at which fibrinogen switches
orientation, from side-on to end-on (S1 to S2), is denoted by k+
3 , and the rate at which fib-
rinogen switches orientation from end-on to side-on (S2 to S1) is denoted by k−
3 .
Applying the law of mass action to reactions (1)-(3), we derive the following pair of ODEs for
the surface concentration of side-on fibrinogen, s1(t), and end-on fibrinogen, s2(t), over time,
t:
ds1
dt
= V k+
1 cs − s1k−
1 − s1k+
3 + s2sk−
3 , (4)
ds2
dt
= V k+
2 cs − s2k−
2 + s1k+
3 − s2sk−
3 , (5)
where c(t) is the concentration of protein in the solution, s(t) is the density of free binding
sites and V is the volume of the solution in which the fibrinogen proteins are suspended. The
initial concentration of fibrinogen, c(0), and the initial density of free binding sites, s(0), are
known. The surface is free of any end-on or side-on orientated fibrinogen initially. Therefore
the following initial conditions can also be imposed:
s1(0) = s2(0) = 0, c(0) = c0, s(0) = sT , (6)
where c0 and sT are known constants. See Tables 1 and 2 for descriptions of the variable and
parameters used in this model.
4

5. Table 1: Variables used in Equations (4)-(13)
Variables Description Units
s1 Density of proteins in the side-on orientation protein molecules m−2
s2 Density of proteins in the end-on orientation protein molecules m−2
c Concentration of protein in the solution protein molecules m−3
s Density of free binding sites binding sites m−2
t Time s
Table 2: Parameters used in Equations (4)-(13)
Parameter Description Units
k+
1 Binding rate of proteins to the surface in the side-on
orientation
binding sites−1
s−1
k−
1 Unbinding rate of proteins from the surface in the
side-on orientation
s−1
k+
2 Binding rate of proteins to the surface in the end-on
orientation
binding sites−1
s−1
k2− Unbinding rate of proteins from the surface in the end-
on orientation
s−1
k3+ Switch rate of proteins on the surface from the side-on
to the end-on orientation
s−1
k3− Switch rate of proteins on the surface from the end-on
to the side-on orientation
binding sites−1
s−1
m2
V Volume of the solution m3
c0 Initial concentration of proteins in the solution protein molecules m−3
A Total surface area of the adsorbent m2
sT Total density of binding sites (both free and occupied) binding sites m−2
µ1(µ2) Number of binding sites occupied by proteins in the
side-on (end-on) orientation
binding sites protein
molecules−1
β Ratio of the surface area of the adsorbent to the vol-
ume of the solution
m−1
5

6. Following [10], we use the conservation of protein and of binding sites to eliminate c and s
from equations (4) and (5) in favour of s1 and s2. By the conservation of mass:
(s1 + s2)dS + cdV = 0, (7)
which simplifies to,
A(s1 + s2) + V c = M, (8)
where M is the number of protein molecules in the solution. Dividing through by V and re-
arranging,
c = c0 − β(s1 + s2), (9)
where c0 is is the initial concentration of proteins in the solution, β = A
V
and c0 = M
V
. For the
conservation of binding sites, we have that:
s = sT − µ1s1 − µ2s2, (10)
where the density of free binding sites, s, is equal to the total density of binding sites, sT ,
minus the density of binding sites occupied by the two protein species, µ1s1 + µ2s2.
Substituting for c and s from Equations (9) and (10) into Equations (4) and (5), we obtain,
ds1
dt
= V k+
1 (c0 − β(s1 + s2))(sT − µ1s1 − µ2s2) − s1k−
1 − s1k+
3 + s2(sT − µ1s1 − µ2s2)k−
3 , (11)
ds2
dt
= V k+
2 (c0 − β(s1 + s2))(sT − µ1s1 − µ2s2) − s2k−
2 + s1k+
3 − s2(sT − µ1s1 − µ2s2)k−
3 , (12)
which depends only upon s1 and s2. The initial conditions have now been reduced to:
s1(0) = s2(0) = 0. (13)
2.1 Nondimensionalisation
In order to reduce Equations (11)-(13) to a dimensionless form, we scale the dependent and
independent variables as follows:
s∗
1 = s1µ2
sT
, s∗
2 = s2µ2
sT
, k+
2
∗
=
k+
2
k+
1
. (14)
Defining the following non-dimensional parameters:
β∗
= sT β
µ2c0
, µ∗
1 = µ1
µ2
, t∗
= tk+
1 V µ2c0, (15)
k−
1
∗
=
k−
1
k+
1 V µ2c0
, k−
2
∗
=
k−
2
k+
1 V µ2c0
, k+
3
∗
=
k+
3
k+
1 V µ2c0
, k−
3
∗
=
sT k−
3
k+
1 V µ2c0
. (16)
Without loss of generality, µ2 can be set as 1, such that a protein which adsorbs onto the sur-
face in orientation s2 occupies one binding site.
Substituting µ2=1 and β = A
V
into β∗
(15), we find that β∗
= AsT
vc0
. Since the initial number
of proteins in the solution, V c0, is much greater than the total number of binding sites, AsT ,
we may simplify the model by setting β∗
=0. In what follows, we shall consider both the cases
6

7. where β∗
=0 and β∗
=1, to examine the effect of protein availability upon the dynamics of the
system.
Dropping the stars, this gives the dimensionless model as,
ds1
dt
=(1 − β(s1 + s2))(1 − µ1s1 − s2) − s1k−
1 − s1k+
3 + s2k−
3 (1 − µ1s1 − s2), (17)
ds2
dt
=k+
2 (1 − β(s1 + s2))(1 − µ1s1 − s2) − s2k−
2 + s1k+
3 − s2k−
3 (1 − µ1s1 − s2), (18)
with initial conditions as in Equation (13).
3 Results - model 1
Using Matlab and the ODE solver, ode15s, Equations (17) and (18) were simulated to show
the concentration of fibrinogen, adsorbed onto the surface, in either the side-on or end-on
orientation.
3.1 Numerical solutions
3.1.1 Case 1: β=1
We begin by setting all dimensionless parameters in equations (17) and (18) to 1, except for
µ1. The parameter µ1 is the ratio of the number of binding sites taken up by fibrinogen in
the side-on orientation, which is 46nm, compared to the end-on orientation, which is 4nm [7].
Therefore, using this ratio, we can take µ1 to be 11.5.
The results in Figure 3 show the fibrinogen in the end-on orientation to adsorb in greater
quantities than in the side-on orientation. This is because the number of binding sites that
the side-on orientation occupies is much more than the number of binding sites that the end-
on orientation occupies.
3.1.2 Case 2: β=0
Comparing the simulations for β=0 and β=1 in Figure 3, it is suggested that for an increased
surface coverage of fibrinogen, the initial supply of fibrinogen molecules has to be greater
than the initial number of binding sites available for adsorption. This can be seen by the
greater s1 and s2 values at t=5 when β=0, compared to the values of s1 and s2 at t=5, for
β=1. (When β=1, the initial number of fibrinogen molecules in the solution is equal to the
number of binding sites available for adsorption.)
3.2 Steady-state analysis
In order to determine the point at which the system is at equilibrium, we perform a steady-
state analysis on equations (17) and (18), with β=1 and β=0.
3.2.1 Case 1: β=1
Taking the case when β=1, we set µ1=11.5 and all other parameters to unity. The only pos-
itive steady-state value produced was (s1,s2)=(0.062,0.15). Since s1 and s2 represent concen-
tration, it is unrealistic to consider any negative steady-state values in the analysis. By con-
sidering the Jacobian matrix, it is shown that the steady-state is a stable node, so the system
7

8. Figure 3: Numerical simulations of the adsorption profile of fibrinogen onto a surface in the
side-on (blue and yellow lines) and the end-on (orange and purple lines) orientation with
µ1=11.5, β=0 or β=1, and all other parameters set to unity. Comparing plots for s1 and s2,
the end-on orientated fibrinogen is found to bind to the surface when β=1 and β=0. Com-
paring β=1 and β=0 for both end-on and side-on orientated proteins, β=0 allows for an over-
all greater surface coverage.
8

9. (a) A phase portrait of model 1, where β=1,
µ1=11.5 and all other parameter values are
set to unity.
(b) A phase portrait of model 1, where β=0,
µ1=11.5 and all other parameter values are
set to unity.
Figure 4: Comparison of phase portraits (a) and (b). Both steady-state values are stable and
it can be seen that when β=0 the steady-state values are slightly greater than the steady-
state value for β=1.
converges towards this fixed point over time. This result can be seen in the phase portrait in
Figure 4 (a).
3.2.2 Case 2: β=0
Considering the case when β=0, the stable node is (s1,s2)=(0.064,0.16). Figure 4 (b) agrees
with the results from Figure 3, that when the initial concentration of fibrinogen in the bulk
solution is large, β=0, the system converges to a larger steady-state value of adsorption for
the end-on and side-on orientation, therefore more fibrinogen adsorbs onto the surface.
Using experimental information from [8], it is said that in order for these orientation changes
to occur when fibrinogen adsorbs onto a surface the initial concentration of the fibrinogen in
the bulk solution must be high. Since this is the behaviour we are trying to model, a parame-
ter value closer to β=0 is more realistic. Therefore, we are going to take β=0 from here on.
Using information from [8] it is seen that in all experiments we start with a completely free
surface, so all binding sites on the surface are available; therefore, Path 1 plotted in Figures 4
(a) and (b) is the most relevant path to the experimental results that we are trying to model.
4 Model 2 formulation: variable reaction rates
One way to improve upon model 1 is to represent the rates of change of fibrinogen’s orien-
tation upon the surface, and the rate of adsorption of fibrinogen in the end-on and side-on
orientation, as modified tanh functions rather than single parameter values. The general form
of this modified tanh function is,
y =
1
2
1 + tanh
x − a
b
, (19)
where the parameters a and b define the switch point and the sharpness of the switch respec-
tively. There is also a translation in the y-direction by 1, and scale by 1
2
so that 0 < y < 1.
9

10. Initially when fibrinogen molecules meet the surface, the side-on orientation is favoured, Equa-
tion (1). This behaviour is due to the surface and fibrinogen being hydrophobic, since the
adsorption of fibrinogen in the side-on orientation covers more binding sites on the surface,
compared to the end-on orientation, bonds are formed between the fibrinogen molecules in
the side-on orientation and the surface, reducing the amount of fibrinogen and surface ex-
posed to water in the solution.
Due to the fact that fibrinogen in the side-on orientation requires more binding sites upon
adsorption, the rate of adsorption of fibrinogen onto the surface in the side-on orientation
is dependent upon the number of free binding sites on the surface. The ability of fibrinogen
molecules to bind to the surface in the side-on orientation is restricted as the number of bind-
ing sites reduce. From this information the switch function for k+
1 is,
ˆk+
1 (s) =
1
2
1 + tanh
(sT − µ1s1 − s2) − ˜s
γ+
1
, (20)
where ˜s represents the centre point at which the rate of the reaction switches to either in-
crease or decrease.
No distinct relationship between fibrinogen already bound to the surface in either orientation
and the rate of desorption of side-on bound fibrinogen from the surface, k−
1 , has be found.
Therefore, it is assumed to be a constant parameter value in model 2, similarly to model 1.
It is also possible, but less likely, that the fibrinogen molecules will bind to the surface in the
end-on orientation in the initial stages of adsorption, Equation (2).
As more fibrinogen molecules start to bind to the surface, the number of free binding sites
reduces; therefore, fibrinogen molecules start to bind more in the end-on orientation as fib-
rinogen binding in this orientation takes up fewer binding sites. Therefore, as s2 increases so
does the binding rate ˆk+
2 . Equivalently, as s2 increases, the binding rate ˆk−
2 reduces: the rate
of the s2 unbinding from the surface will reduce due to the strong bonds formed between the
fibrinogen molecules.
In order for there to be a surge of fibrinogen molecules binding to the surface in the end-on
position, there has to be some fibrinogen already bound to the surface in the end-on position;
therefore, we set φ ≥0 to ensure this. This is because at the initial stages of adsorption there
are more free binding sites available; therefore, the longer length of fibrinogen, side-on ori-
entated, will have more space for adsorption. So in order to activate the surge of end-on ad-
sorption there must be attraction from other end-on fibrinogen already bound on the surface
attracting the free fibrinogen in the solution to the surface in the end-on orientation. This
leads to,
ˆk+
2 (s2) =
k+
2
2
1 + tanh
s2 − η1
γ+
2
+ φ (21)
and
ˆk−
2 (s2) =
k−
2
2
1 − tanh
s2 − η2
γ−
2
, (22)
Once a fibrinogen molecule has bound to the binding sites on the surface it is susceptible to
changing its initial adsorption orientation, Equation (3). The change of orientation from the
side-on to the end-on orientation occurs when there are more end-on fibrinogen molecules
10

11. bound nearby, and the change of orientation from the end-on to side-on orientation occurs
when there are fewer end-on fibrinogen bound nearby.
Proteins bound to binding sites on the surface in the end-on orientation attract nearby fib-
rinogen molecules bound to the surface in the side-on orientation, forcing them to flip up into
the end-on orientation. This suggests that, at steady-state, the end-on orientation will domi-
nate, if not cover, the whole surface.
From the information above we can assume that the change from the side-on to the end-on
orientation only depends upon the value of s2; as s2 increases the rate of this change in orien-
tation increases. Hence,
ˆk+
3 (s2) =
k+
3
2
1 + tanh
s2 − ψ1
γ+
3
, (23)
Similarly, when s2 increases, the rate of the switch from the end-on orientation to side-on
orientation reduces. Hence,
ˆk−
3 (s2) =
k−
3
2
1 − tanh
s2 − ψ2
γ−
3
. (24)
The switch between the side-on and end-on orientation will happen very rapidly as the value
of s2 increases, this is due to the hydrophobic nature of fibrinogen. The fibrinogen covers as
much of its surface as possible so it is exposed to the least amount of solution. Therefore, we
take γ+
3 =γ−
3 =0.1, this is to ensure a sharp gradient around the switch point ˜s2. In Figure 5
ψ1=ψ2=0.1, for the purpose of demonstrating where the switch point occurs.
For each reversible reaction the switch point is the same for the forward and the backward
reaction, as the forward reaction increases the backward reaction will automatically decrease
and vice versa, one does not happen without the other. They may not increase and decrease
at the same rate, but the point at which that change occurs will always be the same. This
behaviour can be seen in Figure 5.
Starting with model 1, and replacing the rate constants k+
2 , k−
2 , k+
3 and k−
3 with the variable
rates defined in Equations (21)-(24) and multiplying the first term in Equation (25) by the
variable reaction rate given by (20), gives us model 2. Due to the nondimensionalisation, k+
1
is not included in the function for ˆk+
1 in Equation (24). This yields:
ds1
dt
=ˆk+
1 (1 − µ1s1 − s2)(1 − β(s1 + s2))(1 − µ1s1 − s2) − s1k−
1 − s1
ˆk+
3 (s2)
+ s2
ˆk−
3 (s2)(1 − µ1s1 − s2), (25)
ds2
dt
=ˆk+
2 (s2)(1 − β(s1 + s2))(1 − µ1s1 − s2) − s2
ˆk−
2 (s2) + s1
ˆk+
3 (s2)
− s2
ˆk−
3 (s2)(1 − µ1s1 − s2). (26)
11

12. Figure 5: Plot of the reaction rates ˆk+
3 and ˆk−
3 . When a switch value of s2 is reached, s2=0.1,
ˆk+
3 starts to increase and ˆk−
3 starts to decrease.
Table 3: Model 2 parameter values
Parameter values
Parameter Case 1 Case 2 Case 3
γ+
2 0.2 0.2 0.2
γ−
2 0.2 0.2 0.2
γ+
1 0.2 0.2 0.2
γ+
3 0.1 0.1 0.01
γ−
3 0.1 0.1 0.2
k−
1 1 1 1
k−
3 1 1 1
k−
2 1 1 1
k+
3 10 10 2
k+
2 10 10 2
µ1 11.5 11.5 11.5
β 0 0 0
φ 0.1 0.1 0.1
ψ1 0.5 0.06 0.06
ψ2 0.5 0.06 0.06
η1 0.5 0.06 0.01
η2 0.5 0.06 0.01
˜s 0.5 0.06 0.01
12

13. Figure 6: The adsorption of fibrinogen of the side-on and end-on orientations for model 2,
using parameter values in Table 3 for case 1, showing s1 to have a greater value than s2. Pa-
rameter values given in Table 3.
5 Results - model 2
5.1 Numerical solutions
5.1.1 Case 1
We begin by simulating the model using the parameter values given by case 2 in Table 3.
Figure 6 shows the final value for s1 to be larger than the final value of s2. Evaluating Fig-
ure 6 it can be seen that no switch between side-on and end-on orientation is occurring here.
This means that only the initial stage of adsorption of fibrinogen is happening, where the
side-on orientation is favoured.
Noting that in Figure 3, the value of s2 is no greater than 0.15, we expect 0.01 ≤ ˜s2 ≤ 0.15;
therefore, this one stage adsorption of fibrinogen is due to the fact that the switch points,
shown in Table 3, are all at 0.5. In fact, the switch point should occur before the steady-state
value is reached. Therefore, the parameter values for the switch points need to be altered so
they produce a graph that shows a second stage of adsorption.
5.1.2 Case 2
To improve on the switch values in case 1, the parameter values ψ1, ψ2, η1, η2 and ˜s were al-
tered to be 0.06, as shown in Table 3, case 2.
Figure 7 shows an initial increase in the value of s1 and then a switch occurs where the steady-
state value for s1 tends to a value close to zero. The end-on orientation tends to a value close
to one, that is, the end-on orientation almost fully covers all binding sites on surface and the
13

14. Figure 7: The adsorption of fibrinogen, and the initial adsorption of fibrinogen inserted, in
the side-on and end-on orientations onto a surface for model 2, using parameter values in Ta-
ble 3 for case 2. Here a change in the rate of adsorption of fibrinogen, in the side-on orienta-
tion, is noticed, going from increasing to decreasing. Parameter values given in Table 3.
14

15. Figure 8: An adsorption profile of fibrinogen using model 2, case 2. The adsorption is shown
by a frequency shift, when adsorption occurs the frequency value reduces. With increasing
protein concentration, a greater frequency shift. Parameter values given in Table 3.
side-on orientation covers almost no binding sites. This switch in the s1 plot in Figure 7 is
because our switch parameter values ψ1, ψ2, η1, η2 and ˜s for model 2, are now 0.06. However,
this initial stage of adsorption in the s2 plot.
Since the aim of the model formulation is to replicate the experimental work published in [8],
an adsorption profile was simulated to compare model 2, case 2, with the explicitly measured
data in Figure 2.
The adsorption of fibrinogen onto the surface is represented by a frequency shift, when ad-
sorption occurs the frequency value will reduce. A sharp drop in the frequency value over a
short period of time represents a fast rate of adsorption. Since model 2 is dimensionless and
the parameter values are estimations, the results simulated from model 2 are not expected to
replicate Figure 2 exactly with respect to the frequency shift values, but the general shape
of the adsorption profile graph should be close to the shape of the adsorption profiles in Fig-
ure 2. Since we are modelling the adsorption of fibrinogen onto a hydrophobic surface, we are
only comparing with the hydrophobic surface plot (circles) in Figure 2.
The shape of the adsorption profile in Figure 8 does not closely replicate the shape of the
adsorption profile in Figure 2. There is no change of gradient in Figure 8 to reflect two differ-
ent stages of adsorption. This suggests that parameter values for case 2 are not an accurate
enough estimation to be used to model the adsorption behaviour of fibrinogen.
5.1.3 Case 3
The adsorption of fibrinogen, in either orientation, will occur first and the unbinding of end-
on orientation closely following. The switch of orientations occurring after the first full ad-
15

16. Figure 9: The adsorption of fibrinogen, and the initial adsorption of fibrinogen (insert), in
the side-on and end-on orientations onto a surface for model 2, using parameter values in Ta-
ble 3 for case 3. There is a change in the rate of adsorption of fibrinogen, going from increas-
ing to decreasing, in the side-on orientation and also a change in rate, going from rapidly in-
creasing to gradually increasing, in adsorption of the end-on orientated fibrinogen.
sorption of fibrinogen, where the rate of the switch from the side-on to the end-on orientated
then happens very rapidly. This suggests that when fibrinogen adsorbs onto a hydrophobic
surface the reactions (1) and (2) occur first, and then reaction (3). Equivalently, in an ex-
periment the rate of each reaction during adsorption is different; therefore, it is unrealistic to
assume that the initial adsorption of end-on and side-on fibrinogen (ˆk+
1 , ˆk+
2 ), the desorption
of end-on and side-on fibrinogen (ˆk−
1 , ˆk−
2 ) and the orientation change (ˆk+
3 , ˆk−
3 ) occur at the
same time and rate during the adsorption process.
Since case 2 did not produce a close enough replication of the adsorption profile, we modify
the parameter estimations γ+
3 , γ−
3 , k+
3 , k−
3 , ψ1, ψ2, η1, η2 and ˜s as in Table 3, case 3.
Comparing Figure 9 with Figure 7, they are similar due to the fact that the end-on orienta-
tion gains almost full coverage of the surface, as it tends to a concentration close to one, and
there are close to no side-on orientated fibrinogen bound on the surface at steady-state. How-
ever the initial stage and second stage of adsorption can be recognised in both s1 and s2 plots
in Figure 9, whereas Figure 7 only showed a two stage adsorption for s1. As fibrinogen has
a two staged adsorption process [8], the parameter values for case 3 are found to provide a
more accurate representation of the behaviour of fibrinogen.
Figure 10 shows an adsorption curve for fibrinogen using model 2, case3, which closely re-
flects the adsorption curve in Figure 2, for the adsorption of fibrinogen onto a hydrophobic
surface produced from experimental results. A change in the rate of adsorption can be seen
by a change in the gradient of the graph in Figure 10, where the fibrinogen molecules are
16

17. Figure 10: An adsorption profile of fibrinogen using model 2, case 3. The adsorption is shown
by a frequency shift, when adsorption occurs the frequency value reduces. With increas-
ing protein concentration there is a greater frequency shift; however, this is seen more so in
the initial stages of adsorption, then the second stage of adsorption. The rate of frequency
change reduces in the second stage of adsorption. Parameter values given in Table 3.
17

18. Figure 11: A derivative plot of the adsorption of fibrinogen using model 2, case 3. The first
stage of adsorption can clearly be seen as the rapid drop in the frequency shift value to -2
between time t=0 and t=0.0003. Parameter values given in Table 3.
changing orientation upon the surface, after the initial stage of adsorption has occurred.
The success of the adsorption profile for model 2, case 3, lead to the simulation of a deriva-
tive plot, as a derivative plot has also been simulated using the experimental results from [8].
The adsorption process for fibrinogen using model 2, case 3, is more clearly visible in Fig-
ure 11, compared to Figure 10, where the initial rapid adsorption is clearly visible, and the
second stage of adsorption is not as visible, but can still be recognised. The adsorption pro-
cess is meant to occur in a stepwise fashion as seen in Figure 2. As the adsorption profile for
model 2, case 3, reflects that in Figure 2, the derivative plot is also expected to reflect that
in Figure 2. However, this is not quite the case here, possible reasons for this which are dis-
cussed in Section 6.
5.2 Steady-state analysis
Steady-state analysis was carried out on model 2, with the three different sets of parameter
values, finding both the number of steady-state values and characterising their stability. Due
to the complicated nature of model 2, the stability of the steady-states were determined by
simulation of the phase portraits.
5.2.1 Case 1
Case 1 exhibits bistability, producing three steady-state values, two of which are stable, as
seen in Figure 12. For high initial values of s2, the solution trajectories tend towards the sta-
18

19. Figure 12: A phase portrait of model 2 using parameter values for case 1 (given in Table
3), showing bistability, where the stable node (s1,s2)≈(0,1) is the desired result and is only
achieved for an initial condition where s2 ≥0.5.
ble node (s1,s2)≈(0,1). However, for low initial values of s2, the trajectories tend towards the
lower stable node where the steady-state value for s2 is lower than the steady-state value for
s1, shown by Figure 12, which is experimentally unrealistic. In experiments carried out on
fibrinogen, the initial concentration of side-on and end-on orientated fibrinogen bound to the
surface always starts at (s1,s2)=(0,0); therefore, the steady-state value of (s1,s2)≈(0,1) will
never be reached using these parameter values.
5.2.2 Case 2
Case 2 has switch point values ψ1, ψ2, η1, η2 and ˜s of 0.06, so that the problems with case 1
are avoided. When the initial condition (s1,s2)=(0,0), the trajectories in the phase portrait
tend towards one stable node (s1,s2)≈(0,1), as shown in Figure 13.
19

20. Figure 13: A phase portrait of model 2 using parameter values for case 2 (given in Table 3),
where there is one stable node at (s1,s2)≈(0,1).
The behaviour of fibrinogen adsorption is found to favour the full coverage of a surface with
end-on orientated proteins only [8].
5.2.3 Case 3
Further improvements were made to switch values and reaction rates, making sure they do
not all occur at the same time, these values are shown in Table 3, case 3, producing a similar
phase portrait to Figure 13.
20

21. Figure 14: A phase portrait of model 2 using parameter values for case 3 (given in Table 3)
showing one stable node at (s1,s2)≈(0,1) and a saddle point.
There are 2 steady-state values in Figure 14, only one of which is stable, (s1,s2)≈(0,1), the
other being a saddle point. No matter what initial conditions are imposed the substrate sur-
face will always end up almost fully covered with end-on orientated fibrinogen, which is what
we would expect to see from fibrinogen [8].
6 Discussion
Looking at the adsorption of fibrinogen through protein concentration plots and conducting
steady-state analyses on models 1 and 2, it is clear that the use of modified tanh functions to
represent variable reaction rates is a more accurate way of reproducing experimental results,
like those seen in [8], compared with constant reaction rates. Similar to the conclusion in [3],
it is inaccurate to represent an orientation change purely on the density of binding sites each
orientation occupies; therefore, the modified tanh functions overcome this by taking into ac-
count the surface chemistry, along with the interactions between the fibrinogen molecules and
the area occupied by the protein in each orientation.
It was suggested that the best value for β would be close to zero, throughout model 2, β was
set to be zero, although any small value, β ≤ 0.5, would still produce similarly shaped ad-
sorption profiles.
It would be assumed that if the adsorption profile of fibrinogen produced by model 2 re-
flected that from [8], that the derivative plots would do also. However, this is not the case.
This may be due to the modified tanh functions in model 2 creating inconsistency in the
derivative plot and the shape of the adsorption profile produced by model 2 not being as
close to the shape of the adsorption profile in Figure 2 as we think.
21

22. To overcome any errors due to the modified tanh functions in model 2, the orientation changes
of fibrinogen could be represented as Hill functions, as demonstrated in [6] and [3]. In [6], a
Hill function for the adsorbed protein concentration was found by equating the ODE for the
reaction to zero, which was then used to deduce the kinetic rate constants for the model. A
Hill function in [3] was made for the mass transport limitation (MTL) of the system, where
‘MTL varies from 0, when the system is not mass transport limited, to 1, when the system is
absolutely mass transport limited’.
In the blood, fibrinogen would be in competition for binding sites on the surface of a bio-
material. Drawing in the model of two compelling proteins (of uniform shape) adsorbing onto
a surface [10], model 2 could be improved by taking into consideration other competing pro-
tein molecules, like albumin, that would more closely replicate the in vivo scenario.
A bio-material’s surface would not be flat when replacing three-dimensional damaged organs
in the body. Therefore, surface curvature needs to be considered, as surface curvature would
affect the orientation of fibrinogen upon adsorption, it’s rearrangement of orientation after
initial adsorption and the overall adsorption profile of fibrinogen.
A further improvement would be to fully parameterise model 2 and investigate how well the
results hold, comparing them with results from [8].
Overall, the modified tanh functions used to represent the reaction rates for fibrinogen of dif-
ferent orientations have been seen to work well when comparing the simulations with experi-
mental results from [8]. However, the factors, such as surface curvature and protein competi-
tion, need to be considered when modelling the complicated nature of protein adsorption on
a larger scale.
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