The Effect of a Linear Tuning between the Antigenic Stimulations of CD4+ T Cells and CD4+ Tregs
<p>Equilibria manifold obtained from Theorem 1. (<b>a</b>) balance between the concentration of T cells <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mi>T</mi> <mo>+</mo> <msup> <mi>T</mi> <mo>*</mo> </msup> </mrow> </semantics></math> and that of Tregs <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mi>R</mi> <mo>+</mo> <msup> <mi>R</mi> <mo>*</mo> </msup> </mrow> </semantics></math>. The shading color indicates the real part of the largest eigenvalue <math display="inline"><semantics> <mrow> <mi>R</mi> <mi>e</mi> <mo>(</mo> <mi>λ</mi> <mo>)</mo> </mrow> </semantics></math>, increasing from black to blue for stable equilibria, and unstable equilibria from green to yellow. The red and the magenta lines show the bifurcations, when <math display="inline"><semantics> <mrow> <mi>R</mi> <mi>e</mi> <mo>(</mo> <mi>λ</mi> <mo>)</mo> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. (<b>b</b>) cross-section for <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0.2765</mn> </mrow> </semantics></math>. The line type indicates stable (solid) or unstable (dashes) equilibria.</p> "> Figure 2
<p>Equilibria manifold obtained from Theorem 1. (<b>a</b>,<b>b</b>) relationship between the antigenic stimulation <span class="html-italic">b</span> of T cells and the concentration of T cells <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mi>T</mi> <mo>+</mo> <msup> <mi>T</mi> <mo>*</mo> </msup> </mrow> </semantics></math>. (<b>c</b>,<b>d</b>) relationship between the antigenic stimulation <span class="html-italic">b</span> of T cells and the concentration of Tregs <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mi>R</mi> <mo>+</mo> <msup> <mi>R</mi> <mo>*</mo> </msup> </mrow> </semantics></math>. (<b>a</b>,<b>c</b>) the axis pointing upwards to the right is the slope parameter <span class="html-italic">m</span>. The shading color indicates the real part of the largest eigenvalue <math display="inline"><semantics> <mrow> <mi>R</mi> <mi>e</mi> <mo>(</mo> <mi>λ</mi> <mo>)</mo> </mrow> </semantics></math>, increasing from black to blue for stable equilibria, and unstable equilibria from green to yellow. The red and the magenta lines show the bifurcations, when <math display="inline"><semantics> <mrow> <mi>R</mi> <mi>e</mi> <mo>(</mo> <mi>λ</mi> <mo>)</mo> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. (<b>b</b>,<b>d</b>) cross-sections for <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0.2765</mn> </mrow> </semantics></math>. The line type indicates stable (solid) or unstalbe (dashes) equilibria.</p> "> Figure 3
<p>Equilibria manifold obtained from Theorem 1. Relationship between the antigenic stimulation <span class="html-italic">b</span> of T cells and (<b>a</b>) the concentration of non-secreting T cells <span class="html-italic">T</span>, (<b>b</b>) the concentration of secreting T cells <math display="inline"><semantics> <msup> <mi>T</mi> <mo>*</mo> </msup> </semantics></math>, (<b>c</b>) the concentration of inactive Tregs <span class="html-italic">R</span>, and (<b>d</b>) the concentration of active Tregs <math display="inline"><semantics> <msup> <mi>R</mi> <mo>*</mo> </msup> </semantics></math>. The axis pointing upwards to the right is the slope parameter <span class="html-italic">m</span>. The shading color indicates the real part of the largest eigenvalue <math display="inline"><semantics> <mrow> <mi>R</mi> <mi>e</mi> <mo>(</mo> <mi>λ</mi> <mo>)</mo> </mrow> </semantics></math>, increasing from black to blue for stable equilibria, and from green to yellow for unstable equilibria. The red and the magenta lines show the bifurcations, when <math display="inline"><semantics> <mrow> <mi>R</mi> <mi>e</mi> <mo>(</mo> <mi>λ</mi> <mo>)</mo> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p> "> Figure 4
<p>Relation between the eigenvalues (<math display="inline"><semantics> <mi>λ</mi> </semantics></math>) with the largest real part (blue line) and the second largest real part (green dashes) with the antigenic stimulation <span class="html-italic">b</span> of T cells, for <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0.2765</mn> </mrow> </semantics></math>. (<b>a</b>) the largest real part of the eigenvalues can be positive for <span class="html-italic">b</span> between <math display="inline"><semantics> <mrow> <msub> <mi>b</mi> <mi>L</mi> </msub> <mo>≈</mo> <mn>2.8</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>b</mi> <mi>H</mi> </msub> <mo>≈</mo> <mn>6.4</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>2</mn> </msup> </mrow> </semantics></math>; and that the second largest real part of the eigenvalues is negative. (<b>b</b>) the two shown eigenvalues can be complex conjugate for <span class="html-italic">b</span> between ∼<math display="inline"><semantics> <mrow> <mn>2.1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math> and ∼<math display="inline"><semantics> <mrow> <mn>2.5</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math>, and for <span class="html-italic">b</span> between ∼<math display="inline"><semantics> <mrow> <mn>1.9</mn> </mrow> </semantics></math> and ∼<math display="inline"><semantics> <mrow> <mn>6.1</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>2</mn> </msup> </mrow> </semantics></math>.</p> "> Figure 5
<p>Time evolutions for two sets of values of the parameters and four initial conditions (see <a href="#mathematics-08-00293-t002" class="html-table">Table 2</a>). (<b>a</b>,<b>b</b>) <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0.2765</mn> </mrow> </semantics></math>. Here, the only stable steady state is the controlled state. (<b>c</b>,<b>d</b>) <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mn>30</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0.2765</mn> </mrow> </semantics></math>. In this case, there are two stable steady states. (<b>a</b>,<b>c</b>) black solid lines—total concentration of T cells <span class="html-italic">x</span>; blue dots—concentration of secreting T cells <math display="inline"><semantics> <msup> <mi>T</mi> <mo>*</mo> </msup> </semantics></math>; green dashes—concentration of non-secreting T cells <span class="html-italic">T</span>. (<b>b</b>,<b>d</b>) black solid lines—total concentration of Tregs <span class="html-italic">y</span>; blue dots—concentration of active Tregs <math display="inline"><semantics> <msup> <mi>R</mi> <mo>*</mo> </msup> </semantics></math>; green dashes—concentration of inactive Tregs <span class="html-italic">R</span>.Time evolutions</p> ">
Abstract
:1. Introduction
2. Theory
3. Equilibria of the Model
4. Stability Analysis
5. Time Evolutions
6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Parameter | Symbol | Range | Value |
---|---|---|---|
T cellT, | |||
T cell Maximum growth rate | day | 4 day | |
Death rate of inactive T cells (day) | 0.1–0.01 [27] | ||
Death rate ratio of active: inactive T cells | 0.01–100 | ||
Capacity of T cells | – cells/ml [28] | cells/ml | |
Input rate of inactive T cells (cells/ml/day) | 0– | 100 | |
Secretion reversion (constant) | k | hrs-days | 0.1 h |
Antigen stimulation level | – | Bifurcation parameter | |
TregsR, | |||
Growth rate ratio T:T | 0.6 | ||
Relaxation rate | hrs-days | 0.1 h | |
Death rate ratio of inactive Tregs: inactive T cells | 1 | ||
Death rate relative ratio of Tregs: T cells | 0.01–100 | 1 | |
Input rate ratio of inactive Tregs: inactive T cells | 1 | ||
Homeostatic capacity | 10– cells/ml | cells/ml | |
Tregs basal antigen stimulation level (for ) | 0–10 per day | 1 per day | |
Homeostatic capacity | 10– cells/ml | cells/ml | |
Secretion inhibition | 0.1–100 | 10 | |
Slope of the tuning | m | 0–1 | Bifurcation parameter |
Cytokines | |||
Max. cytokine concentration | 100–500 pM | 200 pM | |
IL2 secretion rate | 0.07, 2 fgrms h [29] | 10 molecs s cell | |
Cytokine decay rate | hrs-days | 1.5 h [30] |
Initial Condition | |||||
---|---|---|---|---|---|
1: immune response | 30 | 30 | 0 | 200 | |
2: intermediate + | 6 | ||||
3: intermediate - | 5 | ||||
4: controlled | 500 | 500 | 0 | 0 |
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Yusuf, A.A.; Figueiredo, I.P.; Afsar, A.; Burroughs, N.J.; Pinto, A.A.; Oliveira, B.M.P.M. The Effect of a Linear Tuning between the Antigenic Stimulations of CD4+ T Cells and CD4+ Tregs. Mathematics 2020, 8, 293. https://doi.org/10.3390/math8020293
Yusuf AA, Figueiredo IP, Afsar A, Burroughs NJ, Pinto AA, Oliveira BMPM. The Effect of a Linear Tuning between the Antigenic Stimulations of CD4+ T Cells and CD4+ Tregs. Mathematics. 2020; 8(2):293. https://doi.org/10.3390/math8020293
Chicago/Turabian StyleYusuf, Aliyu A., Isabel P. Figueiredo, Atefeh Afsar, Nigel J. Burroughs, Alberto A. Pinto, and Bruno M. P. M. Oliveira. 2020. "The Effect of a Linear Tuning between the Antigenic Stimulations of CD4+ T Cells and CD4+ Tregs" Mathematics 8, no. 2: 293. https://doi.org/10.3390/math8020293