Author’s Accepted Manuscript The intracellular dynamics of hepatitis B virus (HBV) replication with reproduced virion “recycling” Jun Nakabayashi www.elsevier.com/locate/yjtbi

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S0022-5193(16)00098-9 http://dx.doi.org/10.1016/j.jtbi.2016.02.008 YJTBI8537

To appear in: Journal of Theoretical Biology Received date: 19 October 2015 Revised date: 2 February 2016 Accepted date: 5 February 2016 Cite this article as: Jun Nakabayashi, The intracellular dynamics of hepatitis B virus (HBV) replication with reproduced virion “re-cycling”, Journal of Theoretical Biology, http://dx.doi.org/10.1016/j.jtbi.2016.02.008 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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The intracellular dynamics of hepatitis B virus (HBV) replication with reproduced virion re-cycling. Jun Nakabayashi February 17, 2016

1

Abstract

Hepatitis B virus (HBV) is a causative agent of hepatitis.

Clinical outcome of

hepatitis type B depends on the viral titer observed in the peripheral blood of the patient. In the chronic hepatitis patient, production of HBV virion remains low level. On the other hand, the viral load prominently increases in fulminant hepatitis patient as compared with that in the chronic hepatitis patient.

We

previously proposed a mathematical model describing the intracellular dynamics of HBV replication.

Our model claried that there are two distinguishable

replication patterns of HBV named  arrested and  explosive replication.

In

the arrested replication, amount of virion newly reproduced from an infected cell remains low level, while the amount of virion extremely increases in the explosive replication.

Viral load is drastically changed by slight alteration of expression

ratio of 3.5kb RNA to 2.4kb mRNA of HBV.

Though our model provided

the switching mechanism determining the replication pattern of HBV, HBV dynamics is determined by not only the expression pattern of viral genes.

In

this study,  recycling of HBV virion in the replication cycle is investigated as a new factor aecting on the intracellular dynamics of HBV replication.

A part

of newly produced virion of HBV is reused as a core particle that is a resource of HBV replication.

This recycling of HBV virion lowers the threshold for the

explosive replication when waiting time for the next cycle of the replication is large. It is seemingly contradicting that prominent production of HBV is caused by large recycling rate and small release rate of HBV virion from infected cell to extracellular space.

But the recycling of HBV virion can contributes to

the positive feedback cycle of HBV replication for the explosive replication to accumulate the core particle as a resource of HBV replication in an infected cell.

Accumulation of core particle in the infected cell can be risk factor for

the exacerbation of hepatitis rather than rapid release of HBV virion from the infected cell.

1

2

Introduction

Hepatitis B virus (HBV) is a major causative agent of hepatitis. HBV infection is serious global health problem because infectious patients of HBV approaches a third of world's population. Especially, it is estimated that 350 million people chronically infected with HBV [15, 4, 32, 29].

Chronic hepatitis may progress

further severe liver disease such as cirrhosis or hepatocellular carcinoma.

The

infection outcome of HBV is aected by amount of HBV reproduction. The kinetics of HBV in chronic hepatitis is quite dierent from that in acute hepatitis [34, 30, 24].

In the acute hepatitis, the viral titer observed in the peripheral

blood of the patient becomes prominently large.

On the other hand, the viral

load within host remains low level during the long time course of hepatitis in the chronic hepatitis. The drastic change of clinical outcome of HBV infection during the long period of chronic infection is known as are [11, 15, 16, 12, 7]. The viral titer in the patient with are prominently increases as compared with that in the chronic infection of HBV. reproduction is remain unclear.

The cause of such drastic change of HBV

Many mathematical models of HBV infection

are proposed[25, 23, 27, 18, 6, 5]. The dynamics of HBV replication is described as a cell level in these papers.

In most of these models, it is assumed that the

newly produced HBV virion is released from the infected cell with a constant rate.

I proposed a mathematical model describing the intracellular dynamics

of HBV replication [21].

The characteristic of this model is that the detailed

intracellular replication process of HBV is taken into consideration based on molecular biological ndings.

Our model revealed that the intracellular repli-

cation dynamics of HBV can be drastically changed by slight alteration of the viral gene expression pattern.

Two distinguishable replication pattern of HBV

named arrested and explosive were shown in our previous study.

In the ar-

rested replication, the reproduction cycle of HBV is stopped by the exhaustion of core particle and the amount of HBV reproduction in an infected cell then remains a certain positive value.

On the other hand, in the explosive replica-

tion, the exponential reproduction of HBV virion is continued in an infected cell and the amount of newly produced HBV virion prominently increases. Explosive replication is driven by the positive feedback cycle in HBV replication process. Our model provides the theoretical back born of the switching mechanism that drastically increases the production of HBV virion by the transition from arrested to explosive replication. In addition to our theoretical study, the many mutations in core and/or pre-core region of HBV genome increases the expression of 3.5kb RNA are observed in the patient with are or severe liver disease [8, 10, 26, 19, 9]

These studies about HBV mutant support the evolu-

tionary scenario obtained from the simulation based on our model. The intracellular dynamics of HBV replication and clinical outcomes of hepatitis depends not only on the expression pattern of HBV gene. Recently, covalently closed circular DNA (cccDNA) is focused because of its persistence [22, 31]. cccDNA plays a critical role for the replication as a template of viral gene products.

For the chronic infection of HBV, the resource of HBV replication must

be maintained in the infected hepatocytes. It is known that a part of the newly

2

produced virion of HBV is re-used as a resource of second cycle of replication in the life cycle of HBV. This 'recycling' of HBV virion is focused as another factor that aects the intracellular replication dynamics of HBV in addition to the viral gene expression pattern to determine the intracellular dynamics of HBV replication and clinical course of hepatitis in this study.

A part of newly produced

virion is released from an infected hepatocyte through ER-golgi system. The ratio of HBV components in an infected hepatocyte to that in the peripheral blood released from infected hepatocytes is quantitatively investigated [17]. But how the recycling / release ratio of HBV virion contribute to the replication of HBV is poorly understood.

The eect of recycling of HBV virion for the replication

dynamics of HBV is investigated by using a modied version of our mathematical model of HBV infection.

Previously reported model is developed to take

the recycling of HBV virion into consideration. This expanded version of our model reveals the role of recycling of HBV virion determining the replication pattern of HBV and clinical course of HBV hepatitis.

3

A mathematical model of the intracellular replication process of HBV

3.1 Full dynamic model Life cycle of HBV is schematically illustrated in Fig.1.

The intracellular repli-

cation process of HBV is modeled based on molecular biological ndings [20, 33, 2]. Basic scheme of HBV replication included in our model is common with a model previously constructed [21].

In addition to the previous model, the

process that newly produced HBV virion is reused as a core particle is included in a new model of HBV infection. The replication process of HBV starts when a core particle designated by invades into cytoplasm. by the core protein.

yc

Core particle of HBV consist of genome DNA packed

The genome DNA partially double stranded (pdsDNA) is

restored by the viral polymerase packed in the core particle with reaction rate

λ.

This reaction is described as

yc

λ → x

The viral RNAs are transcribed from cccDNA designated by

x.

Two viral

RNAs those are necessary for HBV replication are addressed in this model [28, 3, 1].

First, 3.5kb RNA designated by

genome, polymerase and core protein. designated by

Rs

Rg

is expressed as templates for

On the other hand, 2.4kb mRNA

is expressed as a template for HBV antigen protein known as

HBs antigen. The expressions of these RNAs are described as chemical reaction equation.

3

µc x →

Rg + x

µs →

Rs + x

x Here,

µc

and

µs

indicate expression rate of 3.5kb and 2.4kb RNA, respec-

tively. Polymerase designated by

p

and core protein designated by

lated from 3.5 kb RNA by frame shift with translation rate tively. rate

HBs designated by

βs .

S

βp

and

c are transβc , respec-

is translated from 2.4 kb mRNA with translation

The translation of viral proteins are described as follows:

Rg

βp →

Rg

βc → c + Rg

Rs

βs → S + Rs

p + Rg

Polymerase interacts with 3.5 kb RNA with association rate pregenome RNA-polymerase complex (RNP) designated by

z.

α1

to produce

The genome

DNA is replicated from the 3.5 kb RNA in pregenome RNA-polymerase complex.

α1 →

Rg + p

z

Replicated DNA is packed by a core protein with association rate

α2

to

produce a core particle.

α2 z+c →

yc

A part of newly produced core particle is further packed by HBs antigen with association rate

vc . rate

α3

to produce the complete virion in cytoplasm designated by

Another part of core particle is released to extracellular space with release

γy

without packaging.

4

yc + S

α3 →

yc

γy → yo

vc

A part of newly produced virion in cytoplasm is again used for the replication as a core particle. This is 'recycling' of HBV virion. The core particle in HBV virion is again available for HBV replication. nated by

ρ.

Here, the recycling rate is desig-

Residual part of newly produced virion is released to extracellular

space with release rate

γv

to expand the infection.

vc

ρ → yc

vc

γv → vo

Based on the chemical reaction equation, the time change of the components of HBV is obtained according to law of mass action. A mathematical model of HBV replication with core particle recycling is described as follows:

dx dt dRg dt dRs dt dp dt dc dt dz dt dyc dt dyo dt dS dt

= λyc − δx x = µc x − α1 pRg − δRg Rg = µs x − δRs Rs = βp Rg − α1 pRg − δp p = βc Rg − α2 cz − δc c = α1 pRg − α2 cz − δz z = α2 cz − α3 yc S − λyc − γy yc + ρvc − δyc yc = γy yc − δyo yo = βs Rs − α3 yc S − δS S 5

(1)

dvc dt dvo dt

= α3 yc S − γv vc − ρvc − δvc vc = γv vc − δvo vo

The model is mentioned as a full dynamics model. Notations in this model is summarized in Table 1 by using this model.

The replication dynamics of HBV is investigated

The eect of the 'recycling' infection of HBV virion is

estimated by changing the recycling rate in model 1.

3.2 Replication pattern of HBV The intracellular dynamics of HBV replication is obtained from this model 1.

One HBV core particle invades into cytoplasm as the initial state.

cDNA designated by viral gene products designated by

Rs

The cc-

x is increased by repair the pdsDNA in core particle. The such as 3.5kb RNA designated by Rg and 2.4kb mRNA

are expressed from cccDNA. The concentration of viral gene

products are accumulated and the concentration of virion is then increased. Finally, newly produced virion is released from the infected cell to the extracellular space by exocytosis with release rate

γv .

The time change of the core particle

and virion in the cytoplasm and released to the extracellular space are shown in Fig2A and B. We previously reported that there are two distinguishable replication patterns mentioned as arrested and explosive replication.

Arrested and explosive

replication reect the chronic infection and acute exacerbation of hepatitis, respectively.

In the arrested replication, the core particle is exhausted by pack-

aging to produce the new virion when the expression of 2.4 kb mRNA and HBs protein exceeds a certain threshold level as compared with that of 3.5 kb RNA.

Virion is nitely produced in the arrested replication from the infected

cell as shown Fig2A.

The copy number of cccDNA in the arrested replication

remains several decades. This result coincides with the intrahepatic level of cccDNA obtained from the biopsy samples of chronic hepatitis patients[17]. This result indicates the biological suitability of this model. On the other hand, the virion is explosively reproduced by the positive feedback loop in the replication cycle when 3.5 kb RNA becomes suciently larger than that of 2.4kb RNA. The amount of virion is prominently increased as compared with the arrested replication as shown Fig2B. These replication patterns is reproduced by full model 1 including recycle. The dierence between arrested and explosive replication is shown in Fig2C. The amount of virion converges to a certain positive level after suciently long time has passed from the infection in the arrested replication shown by solid line in Fig2C.

Conversely, the copy number of virion continues

to exponentially increase in the explosive replication shown by dashed line in Fig2C. The amount of newly produced virion from an infected cell is drastically changed between these distinguishable replication patterns. The expression ratio of 3.5kb RNA to 2.4kb RNA designated by critical role determining the replication pattern.

of virion in HBV replication cycle is investigated.

6

µc /µs

plays a

Next, the eect of recycling The relationship between

the expression ratio of 3.5kb RNA to 2.4kb mRNA and the recycling rate of newly produced virion is shown in Fig3A, B, C and D.

In this contour plot,

bright region is indicates the area where the production of HBV virion at a

ρ

certain time point is large. The recycling rate designated by the replication pattern of HBV.

critically aects

The amount of virion always increases in the

upper side of the contour plot of Fig3A, B, C and D.

HBV explosively repli-

cates in the white region in these plots. The eect of recycling rate

ρ

for HBV

virion production depends on time point of the observation. Shortly after from the infection,

vo

ρ becomes small. In the left vo becomes large. After suciently long time vo becomes large on the right hand side in the

increases as the recycling rate

hand side of the contour plot, has passed from the infection,

contour plot as shown in Fig3B, C and D.

Recycling of the virion contributed

to the accumulation of the core particle in the cytoplasm as a source of HBV replication. The positive feedback cycle can function in the white region of the contour plot. as

ρ

The threshold

µc /µs

for the explosive replication becomes small

becomes large after long time has passed from the infection.

Recycling of

HBV virion in replication process lower the threshold of expression ratio for the explosive replication.

3.3 Simplied model For further analysis, full dynamic model is simplied. dation terms are ignored.

First of all, the degra-

If the viral gene derivatives is rapidly degraded as

compare to those of production, newly virion is not accumulated in infected hepatocyte.

So It is proper the assumption that the degradation rate is so small

that degradation terms in the full model 1 are ignored. In the replication cycle of HBV, some components of HBV virion reach quasi-equilibrium when their productions from HBV genome come into balance with their consumptions for the intermediate products. For example, 3.5 kb RNA is expressed to match the interaction with the viral polymerase and its degradation. the time change of 3.5 kb RNA becomes 0.

In this condition,

It is assumed that the compo-

nents of HBV virion immediately reach quasi-equilibrium.

This assumption is

appropriate especially in the early phase of the infection.

According to these

assumptions, the full dynamic model 1 is simplied as follow:

dx dt dRg dt dRS dt dp dt dc dt

=

λyc

=

µc x − α1 pRg

=

µS x

=

βp Rg − α1 pRg

=

βc Rg − α2 cz

7

dz dt dyc dt dyo dt dS dt dvc dt dvo dt

=

α1 pRg − α2 cz

=

α2 cz − α3 yc S − λyc − γy yc + ρvc

=

γy yc

=

βS RS − α3 yc S yc S − γv vc − ρvc

= α3 =

γv vc

Because core particle and virion released to extracellular space cannot aect other viral gene products,

yo

vo can be ignored to consider the intracellular Rg , RS , p, c, z , S and vc reach quasi-equilibrium

and

dynamics of HBV replication.

to balance their consumption by procedure HBV replication.

α1 pRg α3 yc S vc

= α2 cz = (γv + ρ) vc =

= µc x = βS RS βS RS γv + ρ

As a result, the full model 1 is simplied as follows:

dx dt dyc dt

dRs dt

= λyc 

 γv βs Rs γv + ρ   γv βs Rs = µc x − (λ + γy ) yc − γv + ρ = µc x − λyc − γy yc −

(2)

= µs x

The behavior of full model 1 almost consistent with that of simplied model 2 as shown Fig 4A and B. In the arrested replication, cccDNA converges to 0 in the full model 1 indicated by solid line in Fig4A. cccDNA obtained from simplied model 2 indicated by dashed line in Fig4A becomes 0 faster than that from the full model 1. This result represents that the template for the replication is lost when cccDNA becomes 0 in the simplied model 2. As a result, the replication cycle is arrested.

Though cccDNA obtained from simplied model 2 increases

faster as compared with that from full model 1 in the explosive replication as shown in Fig4B, the nal result that cccDNA diverges to innite is coincides between two models.

The systematic behavior of simplied model 2 depends

on cccDNA. It is reported that cccDNA plays important role for the persistent

8

infection of HBV as a source of the replication.

This nation of HBV infection

is well described by simplied model 2. From 2, homogenous equation of

yc is

d2 yc dyc d3 yc + (λ + γy ) 2 − µc λ + 3 dt dt dt



γv γv + ρ

 βs µs λyc

=

0

(3)

Charasteristic equation of 3 is

ω 3 + (λ + γc ) ω 2 − µc λω +



γv γv + ρ

 βs µs λ =

0

(4)

If 4 has three roots, the time dependent solution of 3 is obtained as exponential function as follows:

yc ω1 ,ω2

Here,

= C1 exp (ω1 t) + C2 exp (ω2 t) + C3 exp (ω3 t)

and

ω3

are roots of 4.

C1 , C2 and C3

(5)

are constants. The virion

released to the extracellular space diverges to innite. The discriminant of the characteristic equation 4 is

D

=

4µ3c

 − 27

γv γv + ρ

2

βs2 µ2s λ2

− 4 (λ + γv )

3



γv γv + ρ

 βs µs λ

2

+ (λ + γv ) µ2c λ2 γv / (γv + ρ)

Here,

(6)

in 6 indicates that the reuse rate of intracellular fraction

of newly produced virion as core particle.

The condition of

ρ

that

D

becomes

always positive is

ρ

    2 2 > (−λγv µ2c (λ + γy ) + 4λµc + βs (λ + γy ) γv 2 (λ + γy ) + 9λµc µs  3 p 2 +2 βs2 γv2 (λ + γv ) + 3λµc µ2s )    2 / λµ2c (λ + γy ) + 4λµc (7) ρ satises this condition, characteristic equation 4 has three real roots yc is expressed by exponential and yc then to innite. 7 is condition of ρ for the explosive replication. As shown

When

and the time dependent solution of diverges

in Fig3D, the threshold for the explosive replication obtained from simplied model 7 is plotted. As compared with the threshold pbtained from full model 1, threshold from simplied model 7 is underestimated.

Because the degradation

of viral gene products are ignored in the simplied model 2, the accumulation of viral gene products is excessively estimated. explosive replication is lowered.

As a result, threshold for the

In addition, at least one root of charaster-

istic equation 4 has positive real part, time dependent solution exponentially

9

Algorithm 1

Evolutionary simulation

(1) Set HBV particle with various

n o (i) (i) (i) (i) µc,n , µS,n , γv,n , ρn

n = 1, 2, · · · , N number of generation j = 1, 2, · · · J number of HBV strain   (j) (j) (j) (j) Calculate v (τ ) = f τ, µc,n , µS,n , γv,n , ρn from

(2) For (3) For (4)

(5) Resample HBV particle Prob(k (6) End for

= j) v

(j) vn

(τ ) /

full model 1

P

(j)

vn (k = 1, 2, · · · J)

j

(j) (7) Set µc,n

(i)

(i)

(i)

(i)

(i)

(i)

(i)

= µc,n + ω, µS,n = µS,n + ω, γv,n = γc,n + ω, ρn = ρn + ω, ω :randomnoise ω ∼ N ormal (0, 0.1) (8) End for n

grow.

For these reasons, the threshold obtained from simplied model 2 can-

not be completely coincide with that from full model 1. But the threshold from simplied model 7 is generally coincides with that from full model 1. Especially, the result that threshold expression ratio

ρ

4

µc /µs becomes

small as recycling rate

becomes large is observed in both model.

Within host evolution of recycling

As mentioned above, not only expression ratio of HBV gene but also recycling rate of HBV virion aects the intracellular dynamics of HBV replication.

The

production of virion in an infected cell is drastically changed to exceeds the threshold for the explosive replication.

The clinical course of hepatitis closely

relates to the amount of HBV reproduction in an infected cell.

We previously

proposed the switching scenario from chronic hepatitis where the viral titer in a patient remains low level to exacerbation where production of virion becomes extremely large during the long clinical course of chronic hepatitis.

Explosive

type of HBV is emerged by mutation in core promoter that activates 3.5kb RNA expression through within host evolution.

New exacerbation scenario of

exacerbation of HBV infection including the recycling of HBV virion is proposed. The evolutionary simulation is performed through the following procedure. Many parameter sets as HBV species by drawing the random values.

The amount of

newly produced virion is calculated from full model1 with each parameter value and then the frequency of each HBV strains are determined by sampling importance resampling. ratio of

(j)

vn (τ )

to

PJ

The frequency of HBV strain

(k)

k=1 vn (τ ).

much increases its frequency.

j

is determined by the

Thus HBV strain that reproduced its progeny

Each parameter is renewed adding the random

value. The process of the evolutionary simulation is iterated over and over. After suciently many iterations, HBV strain with the parameter set maximizing the production of virion is selected.

Four parameters,

µc , µs

,

ρ

and

γv

are

handled as free parameters. Sample paths are shown in Fig5A, B and C.

10

Evolutionary simulation is

performed with various xed waiting time for the next cycle of HBV replication designated by (τ

= 20[min]),

τ.

As shown in Fig5A, when the waiting time

τ

is small

the amount of HBV virion released to extracellular space be-

comes large through within host evolution as

ρ

indicating recycling rate and

indicating release rate of HBV virion is small and large , respectively. Both and

µs

γv µc

indicating the expression rate of 3.5kb RNA and 2.4kb mRNA increase

through the evolutionary process. The threshold expression ratio for the explosive replication is not exceeded after long generation time has passed because

ρ

newly produced virion is rapidly released with small

and large

γv

from the

infected cell. The accumulation of core particle as a source of HBV replication is not sucient to introduce the positive feedback cycle of replication. As a result, the amount of virion released to extracellular space gradually increases.

The

evolutonary simulations are performed several times and it is conrmed that almost same results are obtained. As the waiting time for the next cycle of replication becomes large ( τ

ρ

gradually decreased and the dierence between

ρ

and

γv

= 80[min]),

becomes small as

shown in Fig5B. Under the condition that τ becomes suciently large (τ = 160[min]), explosive replication quickly occurs with large expression ratio µc /µs and large recycling rate ρ as shown in Fig5C. The concentration of released virion prominently increases. It is seemingly inconsistent that the released virion increases despite the small release rate of virion

γv .

It is because that the ex-

plosive replication achieved by the sucient accumulation of core particle of HBV as a source of the viral gene products according to the recycling with large

ρ.

ρ.

As shown in Fig3, threshold expression ratio

µc /µs

is lowed by large

Finally, HBV strain that achieve the threshold for the explosive replication

emerges through the evolutionary process and the amount of virion in extracellular space prominently increases.

This result indicates that the exacerbation

is caused by the mutation aecting the release rate of HBV virion from an infected cell through within host evolution during the long clinical course of chronic hepatitis.

5

Discussion

Further to our previous study, the intracellular replication process is modeled based on the molecular biological ndings.

In addition to the expression pat-

tern of viral gene, 'recycling' of HBV virion that is newly produced is taken into consideration.

Our new model reveals that recycling of HBV virion can

contribute to determine the intracellular replication pattern.

Waiting time for

the next cycle of infection becomes small, recycling of HBV virion cannot be appropriate for the ecient expansion of infection.

In this case, HBV strain

with small recycling rate and large release rate of virion becomes optimum.

It

is reported that the intrahepatocytic level of cccDNA is smaller than the serum HBV DNA level in the chronic hepatitis patient [17].

The serum HBV DNA

may reects the amount of HBV virion released to the extracelluar space. This result suggests that the release of newly produced virion is given priority to

11

recycling in the steady state of the chronic hepatitis.

This is suitable for the

arrested replication in our model. As waiting time for the next step of infection becomes large, recycling contributes to the accumulation of core particle in cytoplasm of infected cell.

cc-

cDNA from the recycled core particle can function as the source of the viral gene products. Finally, HBV virion is exponentially reproduced by the positive feedback cycle of replication. source of HBV replication [31].

It is known that cccDNA can function as the The accumulation of the template of the viral

gene is caused by cccDNA introduced from the recycled core particle. Some mutant of HBs protein of HBV obtained from the fulminant hepatitis patient are reported [14, 13]. Impairment of virion secretion is caused by these mutations.

The viral titer in peripheral blood of the fulminant hepatitis pa-

tient becomes extremely large.

It is seemingly contradictory that impairment

of virion release increases the viral load of the hepatitis patient. But it is well explained by our model.

The core particle of HBV with small release rate of

virion is concentrated in the cytoplasm of infected cell.

When waiting time

for the next replication cycle is small, the frequency of mutant strain remains small by competition against wild type HBV strain with larger release rate. As waiting time for the next replication step becomes large, smaller release rate of virion becomes adoptive. The explosive replication is occurred waiting time for the next cycle of infection becomes suciently large. And the positive feedback loop in the replication cycle is then driven by the accumulation of core particle of HBV with small release rate. This is a evolutionary scenario that HBV with the mutation in S-gene that causes fulminant hepatitis. It is reported that the intrahepatic level is much smaller as compared to total serum HBV DNA level [17]. This result indicate that release of newly produced virion is preceding to the accumulation of HBV derived products. This coincides with the case of evolutionary simulation when the waiting time is small. Large release rate of virion designated by

γv

and small recycling rate

for the replication in the chronic hepatitis patient.

ρ

is appropriate

The replication strategy

that small amount of virion is quickly released to expand the next cycle of the replication is advantageous. As mentioned above, our model provides a theoretical back born of the mechanism causing the exacerbation during the chronic HBV infection through within host evolution. The accumulation of virion as a source of replication is risk factor for the exacerbation.

12

Figure 1: Scheme of the intracellular replication process of HBV.

Replication

cycle of HBV has started when core particle invades into cytoplasm. The genome DNA of HBV that is partially double stranded is repaired by its polymerase packed in core particle. DNA (cccDNA).

Viral gene products are expressed from the repaired

3.5kb RNA is expressed as a template for pregenome, core

protein and polymerase.

2.4kb mRNA is expressed as a template for surface

protein. Viral polymerase interacts with 3.5kb RNA to produce the pregenomepolymerase complex (RNP). particle.

RNP is packed by core protein to produce core

Surface protein is translated from 2.4kb mRNA.

Newly produced

core particle is further packed by surface protein to produce HBV virion.

A

part of the produced virion is released to the extracellular space to expand the infection. Another part of virion is reused as core particle for the next cycle of the replication.

13

Figure 2: Time course of cccDNA, the core particle and virion in cytoplasm and extracellular space. mRNA designated by

(A) When expression ratio of 3.5kb RNA to 2.4kb

µc /µs

exhaustion of core particle.

is small, replication cycle of HBV is arrested by In this case, amount of virion remain a certain

positive level. The amount of newly produced virion released to the extracellular space designated

vo is indicated by right and side axis.

(B) When

µc /µs becomes

suciently large, HBV can continue to reproduce its progeny exponentially. (C) Logarithmic plot of virion in extracellular space. There are two distinguishable replication pattern, explosive and arrested indicated by dashed and solid

α1 = 0.1 min-1 , α2 = 0.1 molecules-1 min-1 , α3 = 0.1 molecules min , βp = βc = βs = 0.1 min-1 , λ = 0.1 min-1 , µc = 0.3 -1 -1 -1 -1 -1 or 0.45 min , µs = 0.1 min , ρ = 0.01 min , γyc = 0.1 min , γv = 0.1 min , -1 δx = δy = δRg = δRS = δp = δc = δS = δyc = δyo = δvc = δvo = 0.001 min . line, respectively.

Parameters:

-1

-1

14

Figure 3: dicates large.

Contour plot of virion in extracellular space.

the

region

(A)vo

large as

ρ

(50).

where

the

amount

of

virion

in

White area in-

extracellular

space

is

Shortly after from infection, amount of virion becomes

becomes small.

(B)v o (100).

The area where amount of virion is

large moves to right hand side in the contour plot as a sampling time becomes large.

(C)vo

(150).

The area where explosive replication is occurred emerges

in the right hand side in the contour plot.. the threshold

µc /µs

(D) vo

(200).

As

ρ

becomes large,

for explosive replication becomes small. Threshold for the

explosive replication is lowered by recycling of the virion of HBV. The condition for the explosive replication obtained from 7 is indicated by red line. Tough the threshold is underestimated by the simplied model 2, the threshold obtained from simplied model 2 generally coincides that from the full model 1.

15

Figure 4:

Comparison between full and simplied model.

cccDNA obtained

from full and simplied model are plotted by dashed and solid line.

(A) The

amount of core particle in cytoplasm becomes 0 after long time has passed from the infection when expression ratio

µc /µs

is small.

cccDNA becomes 0

in the simplied model 2, replication cycle arrests when cccDNA converges to 0 in the full model 1.

(B) When expression ratio

µc /µs

is large, cccDNA can

exponentially increase. cccDNA obtained from both full and simplied models diverges to innite.

16

µc , µS , γv

Figure 5: A sample path of evolutionary simulation. change through the evolutionary process.

and

ρ

freely

Evolutionary simulations are per-

formed with various xed waiting time for the next cycle of the replication designated by waiting time

ρ

τ

τ.

Mean value of 1000 HBV strains are plotted.

ρ

is small, HBV strain with small

obviously decreases and then converges to 0.

larger, advantage of small reproduction.

ρ

(B) As waiting time becomes

is lost. Recycling rate

ρ

becomes neutral for HBV

The dierence between recycling rate

newly produced virion further increased.

γv

becomes small.

ρ

and the relese rate of

The expression ratio

(C) Large recycling rate

ρ

(A) When

increases its frequency. Mean

µc /µs

becomes

inversely becomes advantageous

for HBV reproduction as waiting time becomes large.

Finally, the parameter

set achieve the threshold for the explosive replication.

The amount of virion

extremely increases through within host evolution.

17

Table 1: List of the variables and parameters in full model 1

18

Table 1: List of the variables and parameters in full model Notation

Description

x yc yo Rg Rs p c z S vc vo α1 α2 α3 λ βp βc βS γy γv ρ δx δyc δyo δRg δRs δp δc δz δS δv c δvo

cccDNA

?? parameter value

Core particle in cytoplasm Core particle in extracellular space 3.5kb RNA 2.4kb mRNA Polymerase Core protein Pregenome-polymerase complex(RNP) Surface protein Virion in cytoplasm Virion in extracellular space Association rate between pregenome and polymerase Associationrate of RNP and core protein Association rate of core particle and surface protein Reaction rate of DNA repair Translation rate of polymerase Translation rate of core protein Translation rate of surface protein Release rate of core protein Release rate of virion Recycling rate Degradation rate of cccDNA Degradation rate of core particle in cytoplasm Degradation rate of core particle in extracellular space Degradation rate of 3.5kb RNA Degradation rate of 2.4kb RNA Degradation rate of polymerase Degradation rate of core protein Degradation rate of RNP Degradation rate of surface protein Degradation rate of virion in cytoplasm Degradation rate of virion in extracellular space

1

0.1 molecules−1 min−1 0.1 molecules−1 min−1 0.1 molecules−1 min−1 0.1 min−1 0.1 min−1 0.1 min−1 0.1 min−1 0.1 min−1 0.1 min−1

0.01 min−1

0.001 min−1 0.001 min−1 0.001 min−1 0.001 min−1 0.001 min−1 0.001 min−1 0.001 min−1 0.001 min−1 0.001 min−1 0.001 min−1 0.001 min−1

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