ICTS

We are often confronted with chemical reactions in chemistry and biology but several familiar models in physics, such as the zero-range process, can also be modeled as chemical reactions or networks of chemical reactions (CRNs). Mathematical models of CRNs either focus on deterministic models via rate equations (systems of nonlinear ordinary differential equations) which quantify  how the concentrations of the different species change in time, or as  stochastic processes modeled by Master equations.

One of the major theorems pertaining to deterministic models of CRNs is the deficiency zero theorem of Feinberg [1] which lays out conditions (related to a quantity called deficiency) for when a set of coupled non-linear ODE's can have a unique non-zero solution. The surprising aspect of this theorem is that the conditions to be satisfied are completely related to the network itself, while the implications are for the dynamics. In this set of lectures, we will first begin, after introducing CRN's and the terminology,   with explaining the content of this theorem. After this, we will discuss the more recent result of Anderson, Craciun and Kurtz [2], who showed that if the deterministically modeled system satisfies the deficiency zero theorem, then the stochastically modeled version has product-form stationary solutions. After this, time permitting, we can discuss some results and techniques for CRN's which do not have zero deficiency [5].

For understanding the deficiency zero theorem, we will mainly use reference [3] , chapters 16-23. This book also has a number of relevant references  including [4].  For understanding the stochastic version, we will use [2,3].

References:

[1] Martin Feinberg, Lectures on Chemical reaction networks, available for download at https://cbe.osu.edu/chemical-reaction-network-theory

[2]David F. Anderson, Gheorghe Craciun and Thomas G. Kurtz, Bull. Math. Biol. 72 ,1947 (2010).

[3] John Baez and Jacob D. Biamonte , Quantum Techniques in Stochastic Mechanics, World Scientific (2018), also available here https://arxiv.org/pdf/1209.3632.pdf

[4] Jeremy Gunawardena (2003) , Chemical reaction network theory for in-silico biologists, lecture notes http://vcp.med.harvard.edu/papers/crnt.pdf

[5] Supriya Krishnamurthy and Eric Smith, J. Phys. A: Math. Theor. 50, 425002 (2017)

We are often confronted with chemical reactions in chemistry and biology but several familiar models in physics, such as the zero-range process, can also be modeled as chemical reactions or networks of chemical reactions (CRNs). Mathematical models of CRNs either focus on deterministic models via rate equations (systems of nonlinear ordinary differential equations) which quantify  how the concentrations of the different species change in time, or as  stochastic processes modeled by Master equations.

One of the major theorems pertaining to deterministic models of CRNs is the deficiency zero theorem of Feinberg [1] which lays out conditions (related to a quantity called deficiency) for when a set of coupled non-linear ODE's can have a unique non-zero solution. The surprising aspect of this theorem is that the conditions to be satisfied are completely related to the network itself, while the implications are for the dynamics. In this set of lectures, we will first begin, after introducing CRN's and the terminology,   with explaining the content of this theorem. After this, we will discuss the more recent result of Anderson, Craciun and Kurtz [2], who showed that if the deterministically modeled system satisfies the deficiency zero theorem, then the stochastically modeled version has product-form stationary solutions. After this, time permitting, we can discuss some results and techniques for CRN's which do not have zero deficiency [5].

For understanding the deficiency zero theorem, we will mainly use reference [3] , chapters 16-23. This book also has a number of relevant references  including [4].  For understanding the stochastic version, we will use [2,3].

References:

[1] Martin Feinberg, Lectures on Chemical reaction networks, available for download at https://cbe.osu.edu/chemical-reaction-network-theory

[2]David F. Anderson, Gheorghe Craciun and Thomas G. Kurtz, Bull. Math. Biol. 72 ,1947 (2010).

[3] John Baez and Jacob D. Biamonte , Quantum Techniques in Stochastic Mechanics, World Scientific (2018), also available here https://arxiv.org/pdf/1209.3632.pdf

[4] Jeremy Gunawardena (2003) , Chemical reaction network theory for in-silico biologists, lecture notes http://vcp.med.harvard.edu/papers/crnt.pdf

[5] Supriya Krishnamurthy and Eric Smith, J. Phys. A: Math. Theor. 50, 425002 (2017)

We are often confronted with chemical reactions in chemistry and biology but several familiar models in physics, such as the zero-range process, can also be modeled as chemical reactions or networks of chemical reactions (CRNs). Mathematical models of CRNs either focus on deterministic models via rate equations (systems of nonlinear ordinary differential equations) which quantify  how the concentrations of the different species change in time, or as  stochastic processes modeled by Master equations.

One of the major theorems pertaining to deterministic models of CRNs is the deficiency zero theorem of Feinberg [1] which lays out conditions (related to a quantity called deficiency) for when a set of coupled non-linear ODE's can have a unique non-zero solution. The surprising aspect of this theorem is that the conditions to be satisfied are completely related to the network itself, while the implications are for the dynamics. In this set of lectures, we will first begin, after introducing CRN's and the terminology,   with explaining the content of this theorem. After this, we will discuss the more recent result of Anderson, Craciun and Kurtz [2], who showed that if the deterministically modeled system satisfies the deficiency zero theorem, then the stochastically modeled version has product-form stationary solutions. After this, time permitting, we can discuss some results and techniques for CRN's which do not have zero deficiency [5].

For understanding the deficiency zero theorem, we will mainly use reference [3] , chapters 16-23. This book also has a number of relevant references  including [4].  For understanding the stochastic version, we will use [2,3].

References:

[1] Martin Feinberg, Lectures on Chemical reaction networks, available for download at https://cbe.osu.edu/chemical-reaction-network-theory

[2]David F. Anderson, Gheorghe Craciun and Thomas G. Kurtz, Bull. Math. Biol. 72 ,1947 (2010).

[3] John Baez and Jacob D. Biamonte , Quantum Techniques in Stochastic Mechanics, World Scientific (2018), also available here https://arxiv.org/pdf/1209.3632.pdf

[4] Jeremy Gunawardena (2003) , Chemical reaction network theory for in-silico biologists, lecture notes http://vcp.med.harvard.edu/papers/crnt.pdf

[5] Supriya Krishnamurthy and Eric Smith, J. Phys. A: Math. Theor. 50, 425002 (2017)

We are often confronted with chemical reactions in chemistry and biology but several familiar models in physics, such as the zero-range process, can also be modeled as chemical reactions or networks of chemical reactions (CRNs). Mathematical models of CRNs either focus on deterministic models via rate equations (systems of nonlinear ordinary differential equations) which quantify  how the concentrations of the different species change in time, or as  stochastic processes modeled by Master equations.

One of the major theorems pertaining to deterministic models of CRNs is the deficiency zero theorem of Feinberg [1] which lays out conditions (related to a quantity called deficiency) for when a set of coupled non-linear ODE's can have a unique non-zero solution. The surprising aspect of this theorem is that the conditions to be satisfied are completely related to the network itself, while the implications are for the dynamics. In this set of lectures, we will first begin, after introducing CRN's and the terminology,   with explaining the content of this theorem. After this, we will discuss the more recent result of Anderson, Craciun and Kurtz [2], who showed that if the deterministically modeled system satisfies the deficiency zero theorem, then the stochastically modeled version has product-form stationary solutions. After this, time permitting, we can discuss some results and techniques for CRN's which do not have zero deficiency [5].

For understanding the deficiency zero theorem, we will mainly use reference [3] , chapters 16-23. This book also has a number of relevant references  including [4].  For understanding the stochastic version, we will use [2,3].

References:

[1] Martin Feinberg, Lectures on Chemical reaction networks, available for download at https://cbe.osu.edu/chemical-reaction-network-theory

[2]David F. Anderson, Gheorghe Craciun and Thomas G. Kurtz, Bull. Math. Biol. 72 ,1947 (2010).

[3] John Baez and Jacob D. Biamonte , Quantum Techniques in Stochastic Mechanics, World Scientific (2018), also available here https://arxiv.org/pdf/1209.3632.pdf

[4] Jeremy Gunawardena (2003) , Chemical reaction network theory for in-silico biologists, lecture notes http://vcp.med.harvard.edu/papers/crnt.pdf

[5] Supriya Krishnamurthy and Eric Smith, J. Phys. A: Math. Theor. 50, 425002 (2017)

We are often confronted with chemical reactions in chemistry and biology but several familiar models in physics, such as the zero-range process, can also be modeled as chemical reactions or networks of chemical reactions (CRNs). Mathematical models of CRNs either focus on deterministic models via rate equations (systems of nonlinear ordinary differential equations) which quantify  how the concentrations of the different species change in time, or as  stochastic processes modeled by Master equations.

One of the major theorems pertaining to deterministic models of CRNs is the deficiency zero theorem of Feinberg [1] which lays out conditions (related to a quantity called deficiency) for when a set of coupled non-linear ODE's can have a unique non-zero solution. The surprising aspect of this theorem is that the conditions to be satisfied are completely related to the network itself, while the implications are for the dynamics. In this set of lectures, we will first begin, after introducing CRN's and the terminology,   with explaining the content of this theorem. After this, we will discuss the more recent result of Anderson, Craciun and Kurtz [2], who showed that if the deterministically modeled system satisfies the deficiency zero theorem, then the stochastically modeled version has product-form stationary solutions. After this, time permitting, we can discuss some results and techniques for CRN's which do not have zero deficiency [5].

For understanding the deficiency zero theorem, we will mainly use reference [3] , chapters 16-23. This book also has a number of relevant references  including [4].  For understanding the stochastic version, we will use [2,3].

References:

[1] Martin Feinberg, Lectures on Chemical reaction networks, available for download at https://cbe.osu.edu/chemical-reaction-network-theory

[2]David F. Anderson, Gheorghe Craciun and Thomas G. Kurtz, Bull. Math. Biol. 72 ,1947 (2010).

[3] John Baez and Jacob D. Biamonte , Quantum Techniques in Stochastic Mechanics, World Scientific (2018), also available here https://arxiv.org/pdf/1209.3632.pdf

[4] Jeremy Gunawardena (2003) , Chemical reaction network theory for in-silico biologists, lecture notes http://vcp.med.harvard.edu/papers/crnt.pdf

[5] Supriya Krishnamurthy and Eric Smith, J. Phys. A: Math. Theor. 50, 425002 (2017)