Immune System

• Skin• Complement• Immune cells

– Macrophages– T cells– B cellsCytokinesIntro-cellular events

Complement System• 25 Proteins that

complement the activity of antibodies in destroying bacteria– Phagocytosis– Puncturing cell membrane– Proteases cleave proteins

• Rids body of antigen-antibody complexes

• Circulate in blood in in-active form– Creates complement cascade

Phagocytes

• Macrophages– Antigen Presentation– All over

• Dendritic cells– Tenticles used to

present antigen– Located in Spleen

• Neutrophils– Contain granules that

have potent chemicals

T cells

• Mature in Thymus• Regulatory

– Helper T cells– Present antigen to B cells– CD4

• Cytotoxic – CD8

• Both secrete necessary cytokines

• Orchestrate elaborate response

• Memory T cells

MHC Complexes

Two typesBound or Free

Type 1Most cells

Type 2APC

B cells

• Programmed to make one antibody

• Needs APC/Cytokines• Creates Plasma cell

– Factories for antibody

• Done by Geometry

Where do they come from

Cytokines

Cytokine network

Disease of Immune System

• Allergy• Auto Immune Diseases

– Rheumatoid Arthritis– Lupus

• Diabetes• Leukemia• HIV

Memory T cells

Angela Mclean

Memory T Cells• They are antigen-specific T

cells that persist long-term after an infection

• If there is a second encounter with an infection, the memory T cells are reactivated and can reproduce to provide a faster and stronger immune response

The Model and Goals• Uses 5 populations: resting Th cells (W),

activated Th cells (X) memory T cells (M), interleukin 2 (IL-2) (I), and antigen (A).

• Using the model, the population dynamics are illustrated both in vitro and in vivo. In previous models, in vivo and in vitro had the same results, but in experiments it was shown that the two were quite different. This model aims to correct this error.

• The model is created to have no numerical estimates of parameters, so the model’s behavior has all possible types of population behavior.

W X M

A

Immigration from the bone marrow

Native Activated Memory

Antigen

Antigen driven activation

Antigen driven activation

Interleukin-2 driven proliferation

Background activation

IL-2

Equation 1Assume that naïve Th cells migrate from

thymus at a constant rate and naïve cells are activated at a rate proportional to the amount of activated Th cells.

• W = Naïve Th cells• Λ = constant rate of migration• 1/μ = half life of the naïve cells• A = Specific antigen• α = rate of activation of naïve cells

dW/dt = Λ – αAW – μW

Equation 2Assume that:• Proliferation of an activated cell creates two

memory cells – Occurs at a constant rate with high concentration of

activated cells

• The half-life of all Th cells are equal• Memory cells can be reactivated by either

reintroduction of the antigen or from background influences, such as a sequestered antigen

• Memory cells are activated at a faster rate than naïve cells

Equation 2

• X = Activated helper T cells• M = Memory cells• δ = Difference in the rate of activation of

memory cells and the rate of activation of naïve cells

• ε = background activation rate (accounts for random chances that a cell was activated for a different reason)

dX/dt = αAW - ρIX/1+ξX + (δαA+ε)M - μX

Rate of activation

Equation 2

• The rate of change of the activated Th cells is equal to the rate of activation of the naïve cell multiplied by the probability of an antigen and cell binding minus the proliferation rate of the activated Th cells changing to memory cells plus the rate of memory cells reactivating minus the death rate of activated cells.

dX/dt = αAW - ρIX/1+ξX + (δαA+ε)M - μX

Equation 3dM/dt = 2ρIX/1+ξX - (δαA+ε)M - μM

• The rate of change of memory cells is equal to double the rate of proliferation of activated cells (because 2 memory cells are produced) minus the rate of reactivation of memory cells minus the death rate of memory cells.

Equation 4

Assume that:• IL-2 is produced

and absorbed by only activated Th cells

• The half-life of IL-2 is constant

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Equation 4

dI/dt = φX - βIX - ψI

• The rate of change of the amount of IL-2 is equal to the amount of IL-2 made by activated cells minus the amount absorbed by activated Th cells minus the death rate of IL-2.

1/ψ = half-life of IL-2I = amount of IL-2

Equation 5

Assume that:• Activated cells

have a constant growth rate when they are not in the presence of specific immunity

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Equation 5

dA/dt = rA - γAX

• The rate of change of the specific antigen is equal to the growth rate of the antigen in the absence of specific immunity minus the rate of interaction of activated cells and the antigen that causes removal of the antigen.

r = growth rate of antigen

Finding a Steady State• In order to find when the change of the

different populations would be steady, the derivatives are set equal to zero.

• After doing this, it is found that the rates are constant only when A (the amount of antigen) equals zero and when X (the amount of activated Th cells) equals a constant.

• A quadratic equation is derived to find what this constant is. Because it is quadratic, we know two roots will be found or X will be equal to zero.

Finding a Steady State cont

• For a replicating antigen, the only X that can be stable is the positive root of the equation.

• For a non-replicating antigen, X can be the positive root or X can be zero.

• The only time that memory cells will be formed is if X is positive, so we are only interested in the replicating antigen.

• The root will not be a real number unless the background activation rate (ε) is greater than the death rate of the Th cells. This is represented by the fact that when e (e = ε/μ) is less than one, there are no real solutions to the equation.

In Vivo Simulation• This models the changes in the amounts of

memory, naïve, and activated cells in the presence of antigen in vivo (in the body)

• All parameters are the same for each trial, the only difference is the growth rate of the antigen

• At time 0 there was a small amount of replicating antigen was added to the system

• At time 10 there was a large challenging dose of antigen introduced

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In Vivo Simulation contd.

• With an intermediate growth rate the T cells are able to clear out the antigen relatively quickly, and can clear out the infection again much more quickly

• With a fast growth rate, the T cells can’t clear it out completely, and there always is a small amount of the antigen present even after reintroduction of the antigen

In Vivo Simulation contd.• With a slow growth rate a persistent infection is

also established, and the T cells do not clear out the infection because they are only slightly stimulated by the slow-growing antigens. The T cells take a long time to proliferate but when a larger dose of the antigen is reintroduced it is able to completely clear it.

• At the reintroduction, where the amount was equal to the initial amounts in the first two trials, the memory and activated cells are pushed past their threshold, clearing the antigen and returning to a stable state.

In Vitro Simulation• This model is much different because of

three major factors:– There is no chance of random activation– No extra naïve cells come from the bone

marrow– The antigen cannot grow

• The model no longer displays immune memory and a single exposure to antigen leads to a short-lived activation and proliferation.

• All cells convert from naïve to memory

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In Vitro Simulation contd.

• This shows that the amount of activated cells increases because it is in the presence of an antigen, but decreases with IL-2 exposure because the activated cells become memory cells.

• When exogenous IL-2 is added to the system the amount of activated cells decreases at a faster rate.

• The cells convert from being mostly naïve to mostly memory

Conclusions

• In vitro cultures of Th cells must be re-exposed to antigens if they want to maintain proliferation.

• In vivo this achieved through background stimulation (random chance of reactivation)

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Our Conclusions• This new model has achieved its goal, the

distinction between in vivo and in vitro situations. There may be some problems with it, but is so far the best representation of the population dynamics of T helper cells and antigens in the human body and in a culture.

• Possible problems:– In this model, rates including death and growth rates

were assumed to be constant. If the rates were varying, even slightly, there may be a great difference in results.

– In has not been shown that memory cells can hold their memory for as long as the model shows.

– There are many things going on in the body that are unpredictable and impossible to model perfectly.

Basic modelEquation for the dynamics of activated Th cells

Result of modeling T-cell

• A reduced version of the model with just two variables is considered so that isoclines can be inspected.

Result of modeling T-cell-model-

XedA

edA

vXbX

aX

A

qAX −

++−+

+++

+= ]

11

[)1)(1(1

2.

)(.

XcAA −=

0)1()]1())(1[()1( 2 =++−−++++ eXeavbebvXe

Result of modeling T-cell-in vivo simulation-

• First, a small amount of replicating antigen is introduced at time zero, when all cells are present are naïve cells.

• When there is not response of immune system, antigen grow initially.

• Antigen drives naïve cells to become activated and activated cell is divided into two memory cells.

Result of modeling T-cell-in vivo simulation-

• Antigen causes a rise in the number of activated and memory cells.

• The size of the activated and memory population maintained in the absence of the replicating antigen depends only on interactions among immune system.

Result of modeling T-cell-in vitro simulation-

• All is same as earier model in 1990.• There is no cross-reactive stimulation or

antigens.• There is no influx of naive cells.

Conclusion

• T-helper cells need to be re-exposed to antigens every few weeks.

• Immune memory, persistent infection, slow growing persistent infection.

)(.

XcAA −=

XbX

aXAX −

++= 2

2.

1

Conclusion

• Memory cell have some special properties but not long-lived.

• Their ability maintain immune system.

• This model displays memory without invoking long time.