Download 47. Modelling early viral dynamics of FMDV in vivo

Appendix 47
Modelling early viral dynamics of FMDV in vivo
M. Quan1,2*, S. Alexandersen3, L. Matthews2, C. Murphy1, Z. Zhang1, M.E.J. Woolhouse2
1
Institute for Animal Health, Pirbright Laboratory, Pirbright, Woking, Surrey, GU24 ONF, U.K.
2
Centre for Tropical Veterinary Medicine, University of Edinburgh, Easter Bush Veterinary Centre,
Roslin, Midlothian, EH25 9RG, U.K
3
Danish Institute for Food and Veterinary Research, Department of Virology, Lindholm, DK-4771
Kalvehave, Denmark
Abstract:
To quantify and predict virus excretion and infectiousness of animals infected with foot-and-mouth
disease virus (FMDV), an understanding of the virus dynamics in individual animals is needed. In this
paper FMDV dynamics in vivo are described in mathematical terms and possible biological
mechanisms suggested to explain the discrepancies between the mathematical model and the
experimental data. The model described virus concentrations in the circulation and interstitial space,
as well as uninfected and infected epithelial cells. The model was fitted to dose-response curves
obtained from pigs inoculated with tenfold different doses of FMDV O UKG 34/2001 by intravenous
injection (12 pigs in total, 4 pigs per treatment group, 3 treatments). The model qualitatively
reproduced the decrease of FMDV in the circulation seen after intravenous inoculation, followed by an
increase of FMDV due to replication of the virus in epithelial cells. The model output showed much
smaller differences in the timing of viraemia curves between treatment groups than was indicated by
the experimental data. The data were better described by a modified model, in which the rate of
infection of epithelial cells at low FMDV concentrations was limited. The modified model was able to
accurately describe early FMDV dynamics in vivo. Further theoretical and experimental work
suggested by the modelling exercise is discussed.
Introduction:
The causative agent of foot-and-mouth disease (FMD) is the foot-and-mouth disease virus (FMDV), a
positive strand virus that belongs to the Aphthovirus genus, family Picornaviridae (Belsham, 1993).
An important characteristic of FMDV is the highly contagious nature of the virus. Cattle and sheep can
be infected with as little as 10 TCID50 (Gibson and Donaldson, 1986; Donaldson et al., 1987) from as
far as 270 km away from a source of FMDV under exceptional conditions (Gloster et al., 1982).
Considerable effort has been made to quantify the transmission of FMDV experimentally (Donaldson
and Alexandersen, 2001; Aggarwal et al., 2002; Alexandersen et al., 2002a; Alexandersen and
Donaldson, 2002; Alexandersen et al., 2002b; Hughes et al., 2002; Alexandersen et al., 2003) and
describe it in the form of mathematical models (Haydon et al., 1997; Ferguson et al., 2001b, 2001a;
Keeling et al., 2001; Morris et al., 2001). Atmospheric models (Gloster et al., 1981; Donaldson et al.,
1982; Baldock, 1993; Durand and Mahul, 1999; Sorensen et al., 2000; Sorensen et al., 2001; Gloster
and Alexandersen, 2004) have been developed to describe the dissemination of FMDV in the
atmosphere. All the models mentioned are influenced by the infectiousness and amount of virus
excreted by individual animals, yet very little modelling has been undertaken to quantify and predict
the infectiousness of an animal. The objective of this research was to quantify and describe in
mathematical terms the early viral dynamics of FMDV in vivo, in order to predict the excretion of
FMDV and infectiousness of individual animals.
Materials and Methods:
Model A
A model of FMDV replication in vivo (Model A) was considered (Figure 1).
Virus concentrations in the central compartment and interstitial space were represented by X [FMDV
genomes per ml serum (ml-1)] and Y [FMDV genomes per ml serum in the interstitial space (ml-1)]
respectively. FMDV was lost from the interstitial space through 3 routes: drained by the lymphatic
system to re-enter the central compartment at rate kyx, removed (kyo) or used to infect epithelial cells
(β1). FMDV was added to the interstitial space from the central compartment (kxy) or from burst
infected epithelial cells (b).
The total numbers of uninfected and infected epithelial cells were represented by variables C and D
respectively. Infected epithelial cells died at rate kdo and released virus back into the interstitial space
or infected neighbouring uninfected epithelial cells (β2).
This gave rise to the following ordinary differential equations:
dX 1
= k yxY − k xy X
dt λ
(1)
299
dY
= λk xy X − k yxY − k yoY + bk do D
dt
dC
= − β1YC − β 2 kdo DC
dt
dD
= β1YC + β 2 k do DC − k do D
dt
(2)
(3)
(4)
The dimensions and the calculation of the parameters used in the model are described in Table 1.
Model B
In Model B, we explored the effect of limiting the per capita infection rate of epithelial cells by FMDV
(β1) in Model A at low FMDV concentrations, by introducing a sigmoidal term
⎛ Yp ⎞
⎜⎜
⎟
p ⎟
⎝m +Y ⎠
to Equations
3 and 4:
dC
Yp
= − β1
YC − β 2 k do DC
dt
m +Y p
dD
Yp
= β1
YC + β 2 k do DC − k do D
dt
m +Y p
(5)
(6)
The effect of the sigmoidal term on β1 is illustrated in Figure 2.
Results:
A typical output from Model A is shown in Figure 3. A decrease in virus concentration in the central
compartment was followed by an exponential increase. A maximum concentration was reached,
before decreasing when the rate of removal of virus was greater than the rate of viral replication.
The output from Model A was compared to experimental data, as described in Quan et.al. (in press).
In this experiment, blood samples were collected three times a day from pigs inoculated
intravenously with different doses of FMDV (108.1, 107.1 or 106.1 FMDV genomes/animal) (Figure 4).
Model A reproduced the general shape (decrease, followed by an increase of virus concentration in
the central compartment) of a viraemia curve, but showed much smaller differences in the timing of
viraemia curves between treatment groups than was indicated by the experimental data. As a result,
the model did not accurately reproduce the experimental data.
Model B differed from Model A in the modification of β1 by the term
Yp
m +Y p
(Equations 5 & 6). A
typical output from Model B (Figure 5) was similar to the output from Model A (Figure 3), with the
exception that viral growth was delayed in the early stages of infection, an effect of modifying β1.
Parameter m determined the virus concentration of Y β1 reached maximum (Figure 2), the greater
the value of m, the longer the delay in viral growth (results not shown).
The output from Model B was compared to experimental data (Quan et.al., in press) in Figure 6 and a
good overall fit was obtained.
Discussion:
A theoretical framework was developed to explore the early viral dynamics of FMDV in vivo. All the
aspects of the intra-host life cycle of FMDV, from distribution, the infection of epithelial cells,
replication within the cell, cell death with release of progeny virus, infection of neighbouring epithelial
cells were included in Model A. The focus of the model was on viral dynamics in the early stages after
infection and a full description of the adaptive immune response in this model was not included.
Virus concentrations in the central compartment and interstitial space were described. The interstitial
space is an important compartment in the body as it the interface between the central compartment
and the epithelial cells. This space contains water, small solutes, plasma proteins, collagen, elastin,
hydrophilic polymers such as hyaluronate and proteoglycans, fat and cells (Geiger et al., 1984). A
large fraction of the plasma proteins and fluid reservoirs are found in the interstitium of connective
tissues.
300
Experimental work had shown that FMDV in the central compartment decreased rapidly (½ life of 30
min) after intravenous inoculation (unpublished data). The loss of virus from the central compartment
may have been due to the accumulation of the virus in the interstitial space, clearance of FMDV by
the immune system and/or loss of FMDV to epithelial cells. Of the possibilities, the accumulation of
FMDV in the interstitial space was considered the most likely mechanism for the decrease of the virus
from the central compartment.
The immune system as the main mechanism for clearance of FMDV from the central compartment
after an intravenous bolus of FMDV was unlikely based on experimental evidence that FMDV could not
be detected in tissues of the mononuclear-phagocyte system, such as the liver, spleen and lymph
nodes two hours after an intravenous inoculation of 108.1 FMDV genomes (unpublished data). The
detection threshold of the assay used to measure FMDV in tissue was approximately 103.5-4.0 FMDV
genomes/g tissue. If the same inoculation dose was used and we assumed 10% of circulating FMDV
was removed by the liver, the approximate concentration was calculated to be 104.3 FMDV genomes/g
liver in a 25 kg pig [liver weight was calculated as 2.7% live weight (Collin et al., 2001) and a
homogenous concentration of virus in the liver was assumed], above the detection threshold of the
assay. Virus should therefore have been detectable in these tissues if the mononuclear-phagocyte
system was the main mechanism for clearance of FMDV from the central compartment.
The decrease of FMDV from the central compartment may also have been as a result of loss of virus
to epithelial cells. Using this approach, it was not possible to reconcile Model A or other models
(describing virus concentrations in epithelial cells; models and results not shown) with the delays
seen in the experimental data between the time of inoculation and the onset of active viraemia.
Where the uptake of virus by epithelial cells was the main mechanism for the loss of virus from the
central compartment, the models predicted a very quick onset of viraemia (for a wide range of
inoculation doses) and this mechanism was therefore not considered to be the main reason for the
decrease of FMDV from the central compartment.
Model A qualitatively reproduced the general shape of a viraemia curve (Figure 3), but did not
accurately reproduce the dose differences seen in the experimental data, nor capture the delay in
time between inoculation and the start of an active viraemia when virus was undetectable in the
central compartment (eclipse phase) (Figure 4).
A comparison between the output from Model A and the experimental data suggested a non-linear
relationship between inoculation dose and onset of active viraemia. A non-linear relationship was
explored by limiting the rate of infection of epithelial cells at low FMDV concentrations. In vivo
experimental evidence to support this modification is lacking, but a limited infection rate at low FMDV
concentration has been described in vitro (Thorne, 1962). The FMDV type O strain and pig kidney
cells were used to show that the infection rate was proportional to virus concentration at low virus
concentrations. At higher virus concentrations, the infection rate levelled off to a constant value,
resulting in a curve similar to those seen in Figure 2.
Two biological interpretations of a limited rate of infection at low virus concentrations are a minimum
infectious dose per cell hypothesis and virus-virus interactions.
The proportion of cells infected from a given virus concentration can be determined from a poisson
distribution (multiplicity of infection) (Knipe et al., 2001). From a hypothesis in which cells are able to
support viral replication only if they contain a minimum number of viruses, the effective rate of
infection (infection in cells that support replication) can be calculated and shown to be limited at low
virus concentrations. There is no evidence to show that more than one FMDV is needed to initiate a
successful infection in a cell, but the dose dependent inhibition of host protein synthesis and
stimulation of viral protein synthesis by FMDV strongly suggests that a minimum infectious dose is
likely.
Infection of cells with FMDV results in the inhibition of host protein synthesis. The mechanism
involves the cleavage of the translation initiation factor eIF4G by the leader (or L) (Devaney et al.,
1988; Medina et al., 1993) and 3C protease (Belsham et al., 2000) of FMDV. eIF4G is part of the
eIF4F complex that plays a critical role in the recognition of the cap structure of cellular mRNA by the
translation machinery of the cell (Prevot et al., 2003). Translation of FMDV RNA is cap-independent
and depends instead on the presence of an internal ribosome entry site (IRES) element within the 5’
noncoding region (Belsham and Brangwyn, 1990). FMDV infection therefore results in a switch of the
cell’s translational capacity from cellular to viral protein synthesis.
If a viral infection is unable to direct the cell’s translational machinery towards viral protein synthesis,
an abortive infection will be the result. Ohlmann et al., (1995) has shown that the rate of translation
301
of uncapped mRNA in vitro is dose dependent and a determinant of this rate is the concentration of L
protease. At low L protease concentration, the rate of uncapped mRNA translation was directly
proportional to L protease concentration. The greater the number of viruses infecting a cell, the
higher the L protease concentration within the cell and the greater the efficiency in directing the cell’s
translational machinery towards viral protein synthesis. An inability of FMDV to produce viral proteins
at low virus numbers can therefore be expressed as a minimum infectious dose/cell.
It has not been shown for FMDV, but virus-virus interactions can enhance the uptake of virus by
target cells. Non-receptor mediated murine mink cell focus-inducing virus infection could be triggered
in trans by the ecotropic virus glycoprotein expressed on the cell surface in a complex with its
receptor. In addition, trans activation increased the rate of spread of the virus through a population
of target cells, indicating that receptor-dependent and –independent pathways functioned in parallel
(Wensel et al., 2003). The rate of infection of cells by this virus is therefore not a constant and would
increase with increasing virus concentration up to a maximum rate.
The term
Yp
m +Y p
was used to limit β1 at low virus concentrations (Equations 6 & 7). The equation
for calculating the rate of infection was first reported by Michaelis and Menten, (1913) to describe a
saturating reaction rate in enzyme kinetics and has also been used by Holling (1959), to describe
Type III functional response predator-prey interactions.
In the Michaelis-Menten equation, the equivalent to m is the Michaelis constant (Km,). The Km value is
the substrate concentration at which the rate of the reaction is half the maximum rate and indicates
the affinity of an enzyme for substrate. The effect of m on the model was to shift curve a viraemia
curve (solid lines curves in Figures 4 and 6) to the right with increasing values of m. In the context of
Model B, m may be thought of as the affinity of FMDV for the its epithelial cell surface receptor.
The Hill equation is a modification of the Michaelis-Menten equation, in which the substrate
concentration (in this case Y) is raised to a power (p). The modification been used to describe
allosterism, a process in which conformational changes are induced in the binding sites of an enzyme
when a ligand or substrate binds to that enzyme. The conformational change can either increase
(positive cooperativity) or decrease (negative cooperativity) the activity or affinity of the enzyme. In
Model B, increasing values of p amplified dose differences by increasing the separation in time of
viraemia curves of different doses (results not shown).
The fit of Model B to the experimental data was an improvement on Model A as the former
reproduced the dose differences, as well as the eclipse phase in the experimental data, which the
latter model did not.
Conclusions:
• A model to describe the early viral dynamics of FMDV in vivo (Model A) did not accurately
reproduce experimental data (Quan et al., in press).
• Discrepancies between the model and the data could be resolved by limiting the rate of
infection of epithelial cells at low FMDV concentration.
Recommendations:
•
More in vivo quantitative and modelling work should be encouraged to allow for a better
understanding of the detailed quantitative aspects of foot-and-mouth disease.
Acknowledgements:
This work was supported by the Department for Environment, Food and Rural Affairs (DEFRA), UK
and the Biotechnology and Biological Sciences Research Council (BBSRC), UK.
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Table 1
b
Model parameter and variable definitions and calculations.
Number of FMDV genomes released per ml of interstitial space per burst infected cell (ml-1).
A value of 30.6 was calculated from the burst size of FMDV per infected cell ÷ interstitial volume. The
average burst size of an infected cell was taken to be 1×105 virus (S Alexandersen, personal
communication, 2004). The interstitial volume was calculated at 13.09% of body weight (BW) (Pond and
Houpt, 1978), giving a volume of 3 273 ml for a 25 kg pig.
kdo
Per capita rate infected cells die per hour (hour-1).
This rate (0.33) was calculated as the inverse of the cell attached and intracellular life cycle of FMDV in
vitro of between 2.5 and 3.5 h (Cartwright et al., 1957).
kxy
Per capita rate FMDV genomes in central compartment lost to the interstitial space per hour (hour-1).
This rate (1.48) was determined from the mean rate of decrease of FMDV from the serum of pigs UU15 –
UU19 and VC80 – VC83 in samples taken up to 2 h after intravenous inoculation of FMDV (unpublished
data). The rate was calculated from log-linear regression equations fitted through the data.
kyo
Per capita rate FMDV genomes in interstitial space lost per hour (hour-1).
This rate was estimated by fitting the model visually to the experimental data.
kyx
Per capita rate FMDV genomes in interstitial space lost to the central compartment per hour (hour-1).
This rate (0.26) was determined from the mean rate of increase of FMDV in the serum of pigs UQ10 –
UQ21 (Quan et al., in press). Log-linear regression equations were fitted to the steepest part of the
viraemia curves, using a minimum of three data points and over a minimum 24 hour period.
β1
Per capita rate uninfected cells infected per FMDV genome per ml serum in the interstitial space per
hour (ml-1.hour-1).
This rate was estimated by fitting the model visually to the experimental data.
β2
Per capita rate uninfected cells infected per burst infected cell.
This rate was estimated by fitting the model visually to the experimental data.
λ
Ratio of concentration of FMDV genomes in central compartment to concentration in the interstitial
space.
The ratio 0.53:1 was calculated from the volume of plasma (6.91% BW) to interstitial space (13.09%
BW) (Pond and Houpt, 1978).
C
The maximum number of epithelial cells available for infection (C(0)) was estimated from the total
surface body area (BSA) of a pig, calculated from the formula BSA cm2 = 734×(body weight in kg)0.656
(Kelley et al., 1973). We estimated an average cell size of approximately 20 µm3 (Bacha and Bacha,
2000) and a layer of epithelial cells 5 to 10 cells thick susceptible to viral infection. A figure of 1.25 x
1010 susceptible epithelial cells in a 25 kg pig was used in all models.
305
Figure 1 A model of FMDV replication in vivo (Model A). Variables X and Y represents the
concentrations of FMDV in the central compartment and interstitial space; C and D represents
uninfected and infected cells respectively. The parameters (e.g. kxz) are constant per capita rates
β1 = 6x10-12
β1 = 6x10-12, m = 1x101, p = 0.5
β1 = 6x10-12, m = 1x102, p = 0.5
β1 = 6x10-12, m = 1x103, p = 0.5
β1 = 6x10-12, m = 1x104, p = 1
β1
6x10-12
β1 (ml-1.h-1)
5x10-12
Yp
β1
m +Y p
4x10-12
3x10-12
2x10-12
1x10-12
0
100
101
102
103
104
105
106
107
108
Y (ml )
-1
Figure 2 The effect of a sigmoidal term
⎛ Yp ⎞
⎜⎜
⎟
p ⎟
⎝m +Y ⎠
on β1 (see Equations 5 and 6, Model B) where
β1 = the per capita rate of infection of epithelial cells by FMDV and Y = FMDV genomes in the
interstitial space.
306
1010
1010
108
108
106
106
104
104
102
102
100
Epithelial cell numbers
FMDV genomes/ml serum
X - FMDV in the central compartment
Y - FMDV in the interstitial space
C - uninfected epithelial cells
D - infected epithelial cells
100
0
24
48
72
96
Time (hours)
Figure 3 Model A (X(0) = 1×104, kyo = 6.0, β1 = 3x10-11, β2 = 0 and other parameter values
described in
Model output for X, when X(0) = 105
4
Model output for X, when X(0) = 10
Model output for X, when X(0) = 10
3
1010
FMDV genomes/ml serum
109
108
107
106
105
104
103
0
24
48
72
96
120
144
168
192
216
240
Time (hours)
Figure 4 An illustrative parameter fit of Model A (kyo = 6.0, β1 = 3×10-11, β2 = 0 and other
parameter values are described in
Table 1) to experimental data (Quan et. al., in press). Symbol colours indicate different
treatments, i.e. 108.1 (black), 107.1 (grey) and 106.1 (white) FMDV genomes/animal intravenous
inoculation; shapes indicate individual animals. Box plots (using the same symbol colour scheme)
show the median time (and 25th and 75th percentiles) active viraemia was first detected (detection
307
1010
1010
108
108
106
106
104
104
102
102
100
0
24
48
72
96
120
Epithelial cell numbers
FMDV genomes/ml serum
X - FMDV in the central compartment
Y - FMDV in the interstitial space
C - uninfected epithelial cells
D - infected epithelial cells
100
144
Time (hours)
Figure 5 Model B (X(0) = 1×104, kyo = 4.0, β1 = 6×10-12, β2 = 1×10-10, m = 1×102, p = 0.5 and
other parameter values described in
Model output for X, when X(0) = 105
Model output for X, when X(0) = 104
Model output for X, when X(0) = 103
1010
FMDV genomes/ml serum
109
108
107
106
105
104
103
0
24
48
72
96
120
144
168
192
216
240
Time (hours)
Figure 6 An illustrative parameter fit of Model B (kyo = 4.0, β1 = 6×10-12, β2 = 1×10-10, m = 1×102,
p = 0.5 fitted visually to data; other parameter values are described in
Table 1) to experimental data (Quan et al., in press). Symbol colours indicate different treatments,
i.e. 108.1 (black), 107.1 (grey) and 106.1 (white) FMDV genomes/animal intravenous inoculation;
shapes indicate individual animals. Box plots (using the same symbol colour scheme) show the
median time (and 25th and 75th percentile) active viraemia was first detected (detection of FMDV)
2 3
308