11 | Stochastic Reaction Kinetics: HIV modelling

Question 1

Stages of HIV infection?

Answer 1

• 0-3 weeks: primary infection
• 3-9 weeks: acute HIV synd. - wide dissemination of virus, seeding lymphoid organs.
• 9 weeks - 8 years: clinical latency; at ca 7/8 years: constitutional symptoms.
• from ca 7/8 years (without treatment): AIDS, opportunistic diseases, death

Question 2

Is the asymptomatic phase of HIV infection a period of dormancy like with Herpes?

Answer 2

No.
Asymptomatic phase highly dynamic, more than 10e10 virions produced each day for HIV-1.

Question 3

Mechanism of infection - what are the steps?

Answer 3

1. Binding to T-cell CD4 receptor (HIV virion membrane protein recognized)
2. Penetration/uncoating (enters cell, uncoats, releasing proteins including RNA, RT, In, PI.)
3. Reverse transcription (DNA produced from viral RNA by HIV reverse transcriptase)
4. Integration (Viral DNA integrated into DNA of host cell by HIV integrase)
   → infected CD4+T-cell: if activated combat another infection, it produces HI virions instead:
5. Virus production (viral DNA transcribed → RNA → structural proteins / virion RNA)
6. Mat. virion release (budding - use part of cell memb → masked from innate imm. system)

Question 4

HIV vs AIDS?

Answer 4

HIV positive: established virus infection/HIV antibodies

AIDS: HIV pos. & CD4+ < 200 cells/µL

Question 5

Estimated virus half life?

Answer 5

≈ 0.24 days

Question 6

Estimated infected cell half life?

Answer 6

≈ 1.55 days

Question 7

How and by whom was the rate of viral synthesis originally estimated?

Answer 7

Perelson et. al - in latent phase (pre-treatment), viral synthesis and degradation are in equilibrium:

synthesis rate = virion clearance · virion concentration · total fluid volume

Parameter estimates based on PI data available:
- virion clearance
- virion concentration
Physiological estimates:
- total fluid volume

→ 3,1/day · 2·10<sup5</sup>/mL · 16L ≈ 10¹⁰ day

Question 8

What was the interpretation about mutations in Perelson et als 1997 paper?

Answer 8

High probability of mutations:
* HIV genome ≈ 10e4 base pairs
* Reverse transcription error rate ≈ 3 * 10e-5 per base pair

→ Probability of a mutation during reverse transcription = 26%
→ Virtually all viable mutations present
→ Probability of all single-nucleotide mutations occurring on a single day ≈ 100%
→ can lead to relapse

Question 9

What are the three classes of antiretroviral drugs we learnt about?

(2023_1)

Answer 9

• Reverse transcriptase inhibitors (RTI);
• Integrase inhibitors (InI);
• Protease inhibitors (PI).

Question 10

How do RTIs disrupt the life-cycle of HIV?

(2023_1, 2020_1)

Answer 10

Reverse transcriptase inhibitors (RTI): inhibit HIV reverse transcriptase, either by:
- competitively binding as nucleoside analoga (NRTIs)
- or by non-competitively binding to reverse transcriptase at another binding site (NNRTIs).

Question 11

How do InIs disrupt the life-cycle of HIV?

(2023_1, 2020_1)

Answer 11

Integrase inhibitors (InI): inhibit viral enzyme that integrates viral DNA into host DNA.

Question 12

How do PIs disrupt the life-cycle of HIV?

(2023_1, 2020_1)

Answer 12

Protease inhibitors (PI): inhibit (HIV-)protease (final assembly of new virions)

Question 13

HIV ODEs without treatment

For T-cells?

Answer 13

• dT_U/dt = λ + p·T_U(1 - T_U/T_m) - δ_TU·T_U - k·V_I·T_U
• dT/dt = k·V_I·T_U - δ_T^*·T^*

Question 14

HIV ODEs without treatment

For virions?

Answer 14

• dV_I/dt = q·N·δ_T*·T^* - CL·V_I - k·V_I·T_U
• dV_NI/dt = (1-q)N·δ_T*·T^* - CL·V_NI

Question 15

HIV Modelling

What are the species involved in the model ?

(2022_1, 2019_1)

Answer 15

T_U : uninfected CD4+ T-helper cell
T^*₁ : infected CD4+ T-helper cell (stage 1)
T^*₂ : infected CD4+ T-helper cell (stage 2)
V_I : infectious virus particle (virion)
V_NI : non-infectious virus particle (virion)

[concentration] or [copy number]

Question 16

HIV Modelling

What is the difference between stage 1 and stage 2 infected T-cells?

Answer 16

• Need this to model treatment with an integrase inhibitor.
• Stage 1 cells have been penetrated by a virus particle and it has uncoated, releasing its RNA and enzymes including HIV-RT which then produces viral DNA.
• Stage 2 cells: HIV-Integrase has been able to insert the viral DNA into the host DNA, the cell is fully infected.

Question 17

HIV modelling

What does the following parameter represent and what unit does it have?

λ

Answer 17

λ: Production rate of T cells

Question 18

HIV modelling

What does the following parameter represent and what unit does it have?

p

Answer 18

p: Proliferation rate of T cells

Question 19

HIV modelling

What do the following parameters represent and what unit do they have?

T_m

Answer 19

T_m: T cell population limit

Question 20

HIV modelling

What do the following parameters represent and what unit do they have?

δ_TU, δ_T*

Answer 20

δ_TU, δ_T*: Death rates of uninfected and infected T cells

Question 21

HIV modelling

What does the following parameter represent and what unit does it have?

k

Answer 21

k: Infection rate

Question 22

HIV modelling

What does the following parameter represent and what unit does it have?

N

Answer 22

N: Number of virions produced per infected cell

Question 23

HIV modelling

What does the following parameter represent and what unit does it have?

CL

Answer 23

CL: Clearance rate of virus

Question 24

dTU/dt = λ − k · T_U · V</sub>I</sub> − δ_TU · T_U

What is the pre-infection steady state?

(2022_1)

Answer 24

V</sub>I</sub> = 0
(also V</sub>NI</sub>, T1</sub>·T^*)

Solve for T_U

→ T_U = λ / δ_TU

Question 25

HIV treatment modelling

Give the ODEs that need to be amended in order to model treatment with the following class of antiretroviral drug:

integrase inhibitor

Answer 25

We need a “two-stage” models - two types of infected T-Cell.
Effectiveness parameter 0 ≤ η_InI ≤ 1
- dT^*₁/dt = k·V_I·T_U - (1 - η_InI)k_T·T^*₁ - δ_T^*1·T^*₁
- dT^*₂/dt = (1 - η_InI)k_T·T^*₁ - δ_T2·T^*₂

Question 26

HIV treatment modelling

Give the ODEs that need to be amended in order to model treatment with the following class of antiretroviral drug:

Reverse transcriptase inhibitor

Answer 26

effectiveness parameter 0 ≤ η_RT ≤ 1
- dT_U/dt = λ + p·T_U·(1 - T_U/T_m) - δ_TU·T_U - (1 - η_RT)k·V_I·T_U
- dT^*/dt = (1 - η_RT)k·V_I·T_U - δ_T^*·T^*

Question 27

HIV treatment modelling

Give the ODEs that need to be amended in order to model treatment with the following class of antiretroviral drug:

Protease inhibitor

(2022_1, 2019_1)

Answer 27

effectiveness parameter 0 ≤ η_PI ≤ 1
- dV_I/dt = (1 - η_PI)q·N·δ_T^*·T^* - CL·V_I - k·V_I·T_U
- dV_NI/dt = (η_PI·q + (1 - q))·N·δ_T^*·T^* - CL·V_NI

Question 28

Explain why a stochastic formulation is needed to estimate the probability of infection using this model.

(2022_1)

Answer 28

Initial stages of infection: small numbers of viruses and infected cells.
In contrast to a deterministic model, a stochastic model can:
- Represent discrete states such as infection of an individual cell;
- Represent random fluctuations possible at low concentrations;
- Estimate the probability of extinction. (det result: infection or not)

Question 29

HIV modelling

How does one arrive at reasonable initial conditions to model the use of medication?

(2023_1)

Answer 29

• Start with model of untreated infection. (lit values /clinical data for initial species conc)
• Simulate model until it reaches a steady state (= latency)
• Use this steady state as the initial condition for introducing medication.

Question 30

HIV treatment modelling

Describe the curves [of virus copy number or of T1/T2 cells] for treatment with the different classes of drugs

(2020_2 - VI, NI , 2020_2 - T1 and T2 curves of PI and InI were given)

Answer 30

For VI and VNI combined:
- InI: the virion numbers drop the quickest
- RTI and PI have the same curve, virion numbers decrease more gradually than InI but eventually meet after about 15 days

For separate VI:
- PI drops very sharply
- InI in the middle
- RTI most gentle decrease

For separate VNI
- PI increases very sharply after start of treatment then gradually decreeases
- RTi decreases gradually
- Ini decreases quickest

Question 31

HIV treatment modelling

[ on exam graphics were shown]

Consider two graphics of T-cells against time, one for stage T1 infected cells starting at T1(0) (start of treatment), one for stage T2 infected cells starting at T2(0). Each graphic has a red curve and a blue curve.

Figure 1a for T1 infected cells (0-200 days):
- red curve increases sharply from T1(0), peaks and then gradually decreases, approaching 0 after ca 200 days
- blue curve drops sharply and is approximately at 0 within 20 days

Figure 1b forT2 infected cells (0-10 days):
- red curve starts decreasing immediately and close to 0 already within 4/5 days
- blue curve is on a plateau for a few days and then slowly and gradually decreases toward 0, almost there after 10 days

Figure 1a and Figure 1b show the decline of T∗1 cells and T∗2 cells after the intake of different anti-HIV drugs, respectively. This data was simulated with the two-stage HIV model introduced in the lecture [ODEs given].

These simulations considered a pre-treatment steady state and not the early phase of infection.

Use the two-stage HIV infection model to explain the differences between the two curves in Figure 1a and in Figure 1b. [two stage ODEs given] with non-infected target cells TU , target cells in the first/second stage of infection T∗1 / T∗2 , infectious viruses VI and non-infectious viruses VNI. ηInI ∈ [0, 1] is the effect of an integrase inhibitor, ηPI ∈ [0, 1] is the effect of a protease inhibitor.

For each drug a 100 % efficacy was assumed (η = 1). Identify for each curve (blue, red) which drug class was given (protease inhibitors PI, integrase inhibitors InI).

(2020_2)

Answer 31

red curve: InI
blue curve: PI

Graphic for T1 infected cells (0-200 days):
- red curve increases sharply from T1(0), peaks and then gradually decreases, approaching 0 after ca 200 days –> InI. Inhibits T1 becoming T2. This creates a backlog of T1 cells, hence the increase, until the lack of T2 cells to produce virions which can infect TU and make T1 takes effect
- blue curve drops sharply and is approximately at 0 within 20 days –> PI. Less infectious virions (which infect TU to make T1) are produced

Graphic forT2 infected cells (0-10 days):
- red curve starts decreasing immediately and close to 0 already within 4/5 days –> InI. effectivly halts production of T2 cells completely, so the drop is quick.
- blue curve is on a plateau for a few days and then slowly and gradually decreases toward 0, almost there after 10 days –> PI. No new infectious virions produced, however there are still some present which can continue to infect TU cells creating T1 cells, until they die out, hence the short plateau and then slower decrease.

Question 32

Why do PI and RTI treatment have the same curve for virion number / time ?

Answer 32

% -> PIs and RTIs cannot be distinguished this way:
% - for PIs, the infectious-to-non-infectious virus ratio is changed,
% but not the sum of both
% - for RTIs, an early process in the infection cycle is interrupted,
% which leads to a delay in effect (there are still T2 cells
% remaining).
% Both types of treatment lead to a delayed effect determined by the
% same rate-limiting process (the number of T1 cells remaining, which decay quite slowly).