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June  2017, 14(3): 755-775. doi: 10.3934/mbe.2017042

Effects of selection and mutation on epidemiology of X-linked genetic diseases

1. 

Dipartimento di Ingegneria, Università degli Studi del Sannio, Benevento (Italy), Piazza Roma, 21, Benevento 82022, Italy

2. 

School of Electrical and Computer Engineering, College of Engineering, University of Tehran, Tehran, Iran

3. 

School of Aeronautics and Astronautics, Purdue University, West Lafayette, Indiana, USA

* Corresponding author: fverrilli@unisannio.it

Received  March 02, 2016 Accepted  November 11, 2016 Published  December 2016

The epidemiology of X-linked recessive diseases, a class of genetic disorders, is modeled with a discrete-time, structured, non linear mathematical system. The model accounts for both de novo mutations (i.e., affected sibling born to unaffected parents) and selection (i.e., distinct fitness rates depending on individual's health conditions). Assuming that the population is constant over generations and relying on Lyapunov theory we found the domain of attraction of model's equilibrium point and studied the convergence properties of the degenerate equilibrium where only affected individuals survive. Examples of applications of the proposed model to two among the most common X-linked recessive diseases (namely the red and green color blindness and the Hemophilia A) are described.

Citation: Francesca Verrilli, Hamed Kebriaei, Luigi Glielmo, Martin Corless, Carmen Del Vecchio. Effects of selection and mutation on epidemiology of X-linked genetic diseases. Mathematical Biosciences & Engineering, 2017, 14 (3) : 755-775. doi: 10.3934/mbe.2017042
References:
[1]

M. AG and D. SS., The Metabolic and Molecular Bases of Inherited Disease, chapter Color vision and its genetic defects, 5955–76, McGraw-Hill, 2001.

[2]

L. Allen and D. Thrasher, The effects of vaccination in an age-dependent model for varicella and herpes zoster, IEEE Transactions on Automatic Control, 43 (1998), 779-789.  doi: 10.1109/9.679018.

[3]

A. Aswani and C. Tomlin, Computer-aided drug discovery for pathway and genetic diseases, in Proc. 49th IEEE Conference on Decision and Control 2010, 2010,4709–4714 doi: 10.1109/CDC.2010.5717302.

[4]

E. August, G. Craciun and H. Koeppl, Finding invariant sets for biological systems using monomial domination, in CDC, 2012,3001–3006 doi: 10.1109/CDC.2012.6426491.

[5]

C. F. Baer, Does mutation rate depend on itself PLoS Biol, 6 (2008), e52. doi: 10.1371/journal.pbio.0060052.

[6]

J. BeckerR. SchwaabA. Möller-TaubeU. SchwaabW. SchmidtH. H. BrackmannT. GrimmK. Olek and J. Oldenburg, Characterization of the factor viii defect in 147 patients with sporadic hemophilia a: Family studies indicate a mutation type-dependent sex ratio of mutation frequencies, American Journal of Human Genetics, 58 (1996), 657-670. 

[7]

D. Bick, Engineering in genomics-genetic disease diagnosis: Challenges and opportunities, IEEE Engineering in Medicine and Biology Magazine, 14 (1995), 226-228.  doi: 10.1109/51.376771.

[8]

F. BlanchiniE. Franco and G. Giordano, Determining the structural properties of a class of biological models, CDC, 2 (2012), 5505-5510.  doi: 10.1109/CDC.2012.6427037.

[9]

D. J. Bowen, Haemophilia a and haemophilia b: Molecular insights, Molecular Pathology, 55 (2002), 1-18. 

[10]

C. Cannings, Equilibrium, convergence and stability at a sex-linked locus under natural selection, Genetics, 56 (1967), 613-618. 

[11]

J. ChunyanJ. DaqingY. Q. Yang and S. Ningzhong, Dynamics of a multigroup SIR epidemic model with stochastic perturbation, Automatica, 48 (2012), 121-131.  doi: 10.1016/j.automatica.2011.09.044.

[12]

J. F. Crow, How much do we know about spontaneous human mutation rates?, Environmental and Molecular Mutagenesis, 21 (1993), 122–129, URL http://dx.doi.org/10.1002/em.2850210205. doi: 10.1002/em.2850210205.

[13]

J. F. Crow, The origins, patterns and implications of human spontaneous mutation, Nature Reviews Genetics, 1 (2000), 40-47.  doi: 10.1038/35049558.

[14]

C. Del Vecchio, L. Glielmo and M. Corless, Equilibrium and stability analysis of x-chromosome linked recessive diseases model, in Proc. IEEE 51st Annual Conference on Decision and Control (CDC), 2012,4936–4941 doi: 10.1109/CDC.2012.6426443.

[15]

C. Del Vecchio, L. Glielmo and M. Corless, Non linear discrete time epidemiological model for x-linked recessive diseases in 22nd Mediterranean Conference on Control and Automation, June 2014, Palermo, Italy 2014. doi: 10.1109/MED.2014.6961588.

[16]

A. W. F. Edwards, Foundations of Mathematical Genetics Cambridge University Press, 2000.

[17]

W. J. Ewens, Mathematical Population Genetics Ⅰ: Theoretical Introduction, Springer, 2004. doi: 10.1007/978-0-387-21822-9.

[18]

D. P. Germain, General aspects of x-linked diseases, in Fabry Disease: Perspectives from 5 Years of FOS, Oxford PharmaGenesis, 2006.

[19]

H. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.

[20]

http://www.istat.it/it/charts/popolazioneresidente, Authors' last visit on February 2015.

[21]

http://www.who.int/genomics/public/geneticdiseases/en/index2.html, Authors' last visit on February 2015.

[22]

C. Hyeygjeon and A. Astolfi, Control of HIV infection dynamics, IEEE Control Systems Magazine, 28 (2008), 28-39. 

[23]

B. J., Worldwide prevalence of red-green color deficiency., Journal of the Optical Society of America, 29 (2012), 313-320. 

[24]

M. J. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals Princeton University Press, 2008.

[25]

M. J. Khoury, T. H. Beaty and B. H. Cohen, Fundamentals of Genetic Epidemiology, Oxford University Press, 1993.

[26]

M. Lachowicz and J. Miekisz, From Genetics to Mathematics vol. 79 of Series on Advances in Mathematics for Applied Sciences, World Scientific Publications, 2009. doi: 10.1142/9789812837257.

[27]

K. Lange, Mathematical and Statistical Methods for Genetic Analysis, Second Edition, Springer, New York, 2002. doi: 10.1007/978-0-387-21750-5.

[28]

K. Lange, Calculation of the equilibrium distribution for a deleterious gene by the finite fourier transform, Biometrics, 38 (1982), 79-86.  doi: 10.2307/2530290.

[29]

K. Lange and B. Redelings, Disease gene dynamics in a population isolate, An Introduction to Mathematical Modeling in Physiology, Cell Biology, and Immunology: American Mathematical Society, Short Course, January 8-9,2001, New Orleans, Louisiana, 59 (2002), 119-138.  doi: 10.1090/psapm/059/1944517.

[30]

D. Luenberger, Introduction to Dynamic Systems, John Wiley and Sons, 1979.

[31]

T. Nagylaki, Selection and mutation at an x-linked locus, Annals of Human Genetics, 41 (1977), 241-248.  doi: 10.1111/j.1469-1809.1977.tb01920.x.

[32]

T. Nagylaki, Introduction to Theoretical Population Genetics, Biomathematics, 21. Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-76214-7.

[33]

H. A. Orr, Fitness and its role in evolutionary genetics, Nature reviews. Genetics, 10 (2009), 531-539.  doi: 10.1038/nrg2603.

[34]

R. H. Post, Population differences in red and green color vision deficiency: A review, and a query on selection relaxation, Social Biology, 29 (1982), 299-315. 

[35]

D. B. SaakianZ. Kirakosyan and C.-K. Hu, Diploid biological evolution models with general smooth fitness landscapes and recombination, Physical Review E, 77 (2008), 061907.  doi: 10.1103/PhysRevE.77.061907.

[36]

G. D. SmithS. EbrahimS. LewisA. L. HansellL. J. Palmer and P. R. Burton, Genetic epidemiology and public health: Hope, hype, and future prospects, The Lancet, 366 (2005), 1484-1498.  doi: 10.1016/S0140-6736(05)67601-5.

[37]

A. E. Stark, Determining the frequency of sporadic cases of rare X-linked disorders, Annals of Translational Medicine, 4 (2016).

[38]

B. Sturmfels, Polynomial equations and convex polytopes, The American Mathematical Monthly, 105 (1998), 907-922.  doi: 10.2307/2589283.

[39]

J. M. Szucs, Selection and mutation at a diallelic X-linked locus, Journal of Mathematical Biology, 29 (1991), 587-627.  doi: 10.1007/BF00163915.

[40]

R. M. Winter, Estimation of male to female ratio of mutation rates from carrier-detection tests in x-linked disorders, American Journal of Human Genetics, 32 (1980), 582-588. 

[41]

X. Xia, Estimation of HIV/AIDS parameters, Automatica, 39 (2003), 1983-1988.  doi: 10.1016/S0005-1098(03)00220-6.

[42]

N. Yasuda and K. Kondô, No sex difference in mutations rates of duchenne muscular dystrophy, Journal of Medical Genetics, 17 (1980), 106-111.  doi: 10.1136/jmg.17.2.106.

[43]

L. Yeghiazarian and M. Kaiser, Markov model of sex-linked recessive trait transmission, Mathematical and Computer Modelling, 29 (1999), 71-81.  doi: 10.1016/S0895-7177(99)00063-1.

[44]

M. YousefiA. Datta and E. Dougherty, Optimal intervention in markovian gene regulatory networks with random-length therapeutic response to antitumor drug, Biomedical Engineering, IEEE Transactions on, 60 (2013), 3542-3552.  doi: 10.1109/TBME.2013.2272891.

show all references

References:
[1]

M. AG and D. SS., The Metabolic and Molecular Bases of Inherited Disease, chapter Color vision and its genetic defects, 5955–76, McGraw-Hill, 2001.

[2]

L. Allen and D. Thrasher, The effects of vaccination in an age-dependent model for varicella and herpes zoster, IEEE Transactions on Automatic Control, 43 (1998), 779-789.  doi: 10.1109/9.679018.

[3]

A. Aswani and C. Tomlin, Computer-aided drug discovery for pathway and genetic diseases, in Proc. 49th IEEE Conference on Decision and Control 2010, 2010,4709–4714 doi: 10.1109/CDC.2010.5717302.

[4]

E. August, G. Craciun and H. Koeppl, Finding invariant sets for biological systems using monomial domination, in CDC, 2012,3001–3006 doi: 10.1109/CDC.2012.6426491.

[5]

C. F. Baer, Does mutation rate depend on itself PLoS Biol, 6 (2008), e52. doi: 10.1371/journal.pbio.0060052.

[6]

J. BeckerR. SchwaabA. Möller-TaubeU. SchwaabW. SchmidtH. H. BrackmannT. GrimmK. Olek and J. Oldenburg, Characterization of the factor viii defect in 147 patients with sporadic hemophilia a: Family studies indicate a mutation type-dependent sex ratio of mutation frequencies, American Journal of Human Genetics, 58 (1996), 657-670. 

[7]

D. Bick, Engineering in genomics-genetic disease diagnosis: Challenges and opportunities, IEEE Engineering in Medicine and Biology Magazine, 14 (1995), 226-228.  doi: 10.1109/51.376771.

[8]

F. BlanchiniE. Franco and G. Giordano, Determining the structural properties of a class of biological models, CDC, 2 (2012), 5505-5510.  doi: 10.1109/CDC.2012.6427037.

[9]

D. J. Bowen, Haemophilia a and haemophilia b: Molecular insights, Molecular Pathology, 55 (2002), 1-18. 

[10]

C. Cannings, Equilibrium, convergence and stability at a sex-linked locus under natural selection, Genetics, 56 (1967), 613-618. 

[11]

J. ChunyanJ. DaqingY. Q. Yang and S. Ningzhong, Dynamics of a multigroup SIR epidemic model with stochastic perturbation, Automatica, 48 (2012), 121-131.  doi: 10.1016/j.automatica.2011.09.044.

[12]

J. F. Crow, How much do we know about spontaneous human mutation rates?, Environmental and Molecular Mutagenesis, 21 (1993), 122–129, URL http://dx.doi.org/10.1002/em.2850210205. doi: 10.1002/em.2850210205.

[13]

J. F. Crow, The origins, patterns and implications of human spontaneous mutation, Nature Reviews Genetics, 1 (2000), 40-47.  doi: 10.1038/35049558.

[14]

C. Del Vecchio, L. Glielmo and M. Corless, Equilibrium and stability analysis of x-chromosome linked recessive diseases model, in Proc. IEEE 51st Annual Conference on Decision and Control (CDC), 2012,4936–4941 doi: 10.1109/CDC.2012.6426443.

[15]

C. Del Vecchio, L. Glielmo and M. Corless, Non linear discrete time epidemiological model for x-linked recessive diseases in 22nd Mediterranean Conference on Control and Automation, June 2014, Palermo, Italy 2014. doi: 10.1109/MED.2014.6961588.

[16]

A. W. F. Edwards, Foundations of Mathematical Genetics Cambridge University Press, 2000.

[17]

W. J. Ewens, Mathematical Population Genetics Ⅰ: Theoretical Introduction, Springer, 2004. doi: 10.1007/978-0-387-21822-9.

[18]

D. P. Germain, General aspects of x-linked diseases, in Fabry Disease: Perspectives from 5 Years of FOS, Oxford PharmaGenesis, 2006.

[19]

H. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.

[21]

http://www.who.int/genomics/public/geneticdiseases/en/index2.html, Authors' last visit on February 2015.

[22]

C. Hyeygjeon and A. Astolfi, Control of HIV infection dynamics, IEEE Control Systems Magazine, 28 (2008), 28-39. 

[23]

B. J., Worldwide prevalence of red-green color deficiency., Journal of the Optical Society of America, 29 (2012), 313-320. 

[24]

M. J. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals Princeton University Press, 2008.

[25]

M. J. Khoury, T. H. Beaty and B. H. Cohen, Fundamentals of Genetic Epidemiology, Oxford University Press, 1993.

[26]

M. Lachowicz and J. Miekisz, From Genetics to Mathematics vol. 79 of Series on Advances in Mathematics for Applied Sciences, World Scientific Publications, 2009. doi: 10.1142/9789812837257.

[27]

K. Lange, Mathematical and Statistical Methods for Genetic Analysis, Second Edition, Springer, New York, 2002. doi: 10.1007/978-0-387-21750-5.

[28]

K. Lange, Calculation of the equilibrium distribution for a deleterious gene by the finite fourier transform, Biometrics, 38 (1982), 79-86.  doi: 10.2307/2530290.

[29]

K. Lange and B. Redelings, Disease gene dynamics in a population isolate, An Introduction to Mathematical Modeling in Physiology, Cell Biology, and Immunology: American Mathematical Society, Short Course, January 8-9,2001, New Orleans, Louisiana, 59 (2002), 119-138.  doi: 10.1090/psapm/059/1944517.

[30]

D. Luenberger, Introduction to Dynamic Systems, John Wiley and Sons, 1979.

[31]

T. Nagylaki, Selection and mutation at an x-linked locus, Annals of Human Genetics, 41 (1977), 241-248.  doi: 10.1111/j.1469-1809.1977.tb01920.x.

[32]

T. Nagylaki, Introduction to Theoretical Population Genetics, Biomathematics, 21. Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-76214-7.

[33]

H. A. Orr, Fitness and its role in evolutionary genetics, Nature reviews. Genetics, 10 (2009), 531-539.  doi: 10.1038/nrg2603.

[34]

R. H. Post, Population differences in red and green color vision deficiency: A review, and a query on selection relaxation, Social Biology, 29 (1982), 299-315. 

[35]

D. B. SaakianZ. Kirakosyan and C.-K. Hu, Diploid biological evolution models with general smooth fitness landscapes and recombination, Physical Review E, 77 (2008), 061907.  doi: 10.1103/PhysRevE.77.061907.

[36]

G. D. SmithS. EbrahimS. LewisA. L. HansellL. J. Palmer and P. R. Burton, Genetic epidemiology and public health: Hope, hype, and future prospects, The Lancet, 366 (2005), 1484-1498.  doi: 10.1016/S0140-6736(05)67601-5.

[37]

A. E. Stark, Determining the frequency of sporadic cases of rare X-linked disorders, Annals of Translational Medicine, 4 (2016).

[38]

B. Sturmfels, Polynomial equations and convex polytopes, The American Mathematical Monthly, 105 (1998), 907-922.  doi: 10.2307/2589283.

[39]

J. M. Szucs, Selection and mutation at a diallelic X-linked locus, Journal of Mathematical Biology, 29 (1991), 587-627.  doi: 10.1007/BF00163915.

[40]

R. M. Winter, Estimation of male to female ratio of mutation rates from carrier-detection tests in x-linked disorders, American Journal of Human Genetics, 32 (1980), 582-588. 

[41]

X. Xia, Estimation of HIV/AIDS parameters, Automatica, 39 (2003), 1983-1988.  doi: 10.1016/S0005-1098(03)00220-6.

[42]

N. Yasuda and K. Kondô, No sex difference in mutations rates of duchenne muscular dystrophy, Journal of Medical Genetics, 17 (1980), 106-111.  doi: 10.1136/jmg.17.2.106.

[43]

L. Yeghiazarian and M. Kaiser, Markov model of sex-linked recessive trait transmission, Mathematical and Computer Modelling, 29 (1999), 71-81.  doi: 10.1016/S0895-7177(99)00063-1.

[44]

M. YousefiA. Datta and E. Dougherty, Optimal intervention in markovian gene regulatory networks with random-length therapeutic response to antitumor drug, Biomedical Engineering, IEEE Transactions on, 60 (2013), 3542-3552.  doi: 10.1109/TBME.2013.2272891.

Figure 1.  Inheritance pattern of X-linked recessive disease
Figure 2.  Region of attraction corresponding to wi parameters in Table 3
Figure 3.  Region of attraction corresponding to wi parameters in Table 3 using the two Lyapunov functions.
Figure 4.  Region of attraction of xB = (72.5, 26.5, 68.9) and system's parameters as in Section 5.2.
Figure 5.  Trajectories with different initial conditions and wi in (27)
Figure 6.  Trajectories converging to xB
Figure 7.  Static sensitivity analysis with respect to wi
Table 1.  X-linked recessive inheritance probabilities for sons
PARENTS SONS
father mother healthy
$X^A Y$
affected
$X^a Y$
$X^A Y$ $X^A X^A$ $1-\gamma$ $\gamma $
$X^A Y$ $X^A X^a$ $\frac{1}{2}(1-\gamma)$ $\frac{1}{2}(1+\gamma) $
$X^A Y$ $X^a X^a$ $0$ $1$
$X^a Y$ $X^A X^A$ $1-\gamma$ $\gamma$
$X^a Y$ $X^A X^a$ $\frac{1}{2}(1-\gamma)$ $\frac{1}{2}(1+\gamma) $
$X^a Y$ $X^a X^a$ $0$ $1$
PARENTS SONS
father mother healthy
$X^A Y$
affected
$X^a Y$
$X^A Y$ $X^A X^A$ $1-\gamma$ $\gamma $
$X^A Y$ $X^A X^a$ $\frac{1}{2}(1-\gamma)$ $\frac{1}{2}(1+\gamma) $
$X^A Y$ $X^a X^a$ $0$ $1$
$X^a Y$ $X^A X^A$ $1-\gamma$ $\gamma$
$X^a Y$ $X^A X^a$ $\frac{1}{2}(1-\gamma)$ $\frac{1}{2}(1+\gamma) $
$X^a Y$ $X^a X^a$ $0$ $1$
Table 2.  X-linked recessive inheritance probabilities for daughters
PARENTS DAUGHTERS
father mother healthy
$X^A X^A$
carrier
$X^A X^a$
affected
$X^a X^a$
$X^A Y$ $X^A X^A$ $(1-\gamma)^2$ $2\gamma(1-\gamma) $ $\gamma^2$
$X^A Y$ $X^A X^a$ $\frac{1}{2}(1-\gamma)^2$ $\frac{1}{2}(1+\gamma-2\gamma^2)$ $\frac{1}{2}\gamma(1+\gamma)$
$X^A Y$ $X^a X^a$ $0$ $1-\gamma$ $\gamma$
$X^a Y$ $X^A X^A$ $0$ $1-\gamma$ $\gamma$
$X^a Y$ $X^A X^a$ $0$ $\frac{1}{2}(1-\gamma)$ $\frac{1}{2}(1+\gamma)$
$X^a Y$ $X^a X^a$ $0$ $0$ $1$
PARENTS DAUGHTERS
father mother healthy
$X^A X^A$
carrier
$X^A X^a$
affected
$X^a X^a$
$X^A Y$ $X^A X^A$ $(1-\gamma)^2$ $2\gamma(1-\gamma) $ $\gamma^2$
$X^A Y$ $X^A X^a$ $\frac{1}{2}(1-\gamma)^2$ $\frac{1}{2}(1+\gamma-2\gamma^2)$ $\frac{1}{2}\gamma(1+\gamma)$
$X^A Y$ $X^a X^a$ $0$ $1-\gamma$ $\gamma$
$X^a Y$ $X^A X^A$ $0$ $1-\gamma$ $\gamma$
$X^a Y$ $X^A X^a$ $0$ $\frac{1}{2}(1-\gamma)$ $\frac{1}{2}(1+\gamma)$
$X^a Y$ $X^a X^a$ $0$ $0$ $1$
Table 3.  Parameters values
N $\gamma$ $w_{13}$ $w_{14}$ $w_{15}$ $w_{23}$ $w_{24}$ $r$
scenario 1 $150$ $10^{-4}$ 0.5 0.45 0.1 1 0.9 11232
scenario 2 150 $10^{-4}$ 0.5 1 0.5 0.5 1 8284
scenario 3 150 $10^{-4}$ 0.63 1.4 0.14 0.7 1.5 2121
N $\gamma$ $w_{13}$ $w_{14}$ $w_{15}$ $w_{23}$ $w_{24}$ $r$
scenario 1 $150$ $10^{-4}$ 0.5 0.45 0.1 1 0.9 11232
scenario 2 150 $10^{-4}$ 0.5 1 0.5 0.5 1 8284
scenario 3 150 $10^{-4}$ 0.63 1.4 0.14 0.7 1.5 2121
Table 4.  Parameters values for Lyapunov function in (14)
$\alpha^*$ $\beta^*$ $\mu^*$ $\underline{x}_1$
scenario 1 0.0917 0.9999 0.9072 150
scenario 2 0.2486 0.3729 0.4953 150
scenario 3 0.0247 0.1359 0.1766 63.6
$\alpha^*$ $\beta^*$ $\mu^*$ $\underline{x}_1$
scenario 1 0.0917 0.9999 0.9072 150
scenario 2 0.2486 0.3729 0.4953 150
scenario 3 0.0247 0.1359 0.1766 63.6
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