Introduction Mass-action Kinetics Negative Numbers Conclusion Chemical Analog Computing with Negative Numbers Hendrik Happe hendrik.happe@uni-jena.de Nineteenth International Conference on Membrane Computing September 11, 2018 Chemical Analog Computing with Negative Numbers Hendrik Happe 1/ 30
Introduction Mass-action Kinetics Negative Numbers Conclusion Introduction Mass-action Kinetics Representation of Negative Numbers Conclusion Chemical Analog Computing with Negative Numbers Hendrik Happe 2/ 30
✌ ✌ ✌ ✌ Introduction Mass-action Kinetics Negative Numbers Conclusion Advantages of chemical computing Chemical Analog Computing with Negative Numbers Hendrik Happe 3/ 30
✌ ✌ ✌ Introduction Mass-action Kinetics Negative Numbers Conclusion Advantages of chemical computing ✌ High storage density Chemical Analog Computing with Negative Numbers Hendrik Happe 3/ 30
✌ ✌ Introduction Mass-action Kinetics Negative Numbers Conclusion Advantages of chemical computing ✌ High storage density ✌ High energy efficient Chemical Analog Computing with Negative Numbers Hendrik Happe 3/ 30
✌ Introduction Mass-action Kinetics Negative Numbers Conclusion Advantages of chemical computing ✌ High storage density ✌ High energy efficient ✌ High parallelism Chemical Analog Computing with Negative Numbers Hendrik Happe 3/ 30
Introduction Mass-action Kinetics Negative Numbers Conclusion Advantages of chemical computing ✌ High storage density ✌ High energy efficient ✌ High parallelism ✌ Recyclable Chemical Analog Computing with Negative Numbers Hendrik Happe 3/ 30
� Ý Ñ ✒ r s ✒ r s ñ ✒ r s ☎ r s ♣r sq ✏ ☎ r s ☎ r s ♣r sq ✏ ✏ ♣r sq ✁ ♣r sq ✏ ☎ r s ☎ r s Introduction Mass-action Kinetics Negative Numbers Conclusion Mass-action Kinetics Assumption: number of effective reactant collisions Z proportional to reactant concentrations (Guldberg 1867) Chemical Analog Computing with Negative Numbers Hendrik Happe 4/ 30
♣r sq ✏ ☎ r s ☎ r s ♣r sq ✏ ✏ ♣r sq ✁ ♣r sq ✏ ☎ r s ☎ r s Introduction Mass-action Kinetics Negative Numbers Conclusion Mass-action Kinetics Assumption: number of effective reactant collisions Z proportional to reactant concentrations (Guldberg 1867) ˆ k A � B Ý Ñ C . . . Z C ✒ r A s and Z C ✒ r B s ñ Z C ✒ r A s ☎ r B s Chemical Analog Computing with Negative Numbers Hendrik Happe 4/ 30
♣r sq ✏ ✏ ♣r sq ✁ ♣r sq ✏ ☎ r s ☎ r s Introduction Mass-action Kinetics Negative Numbers Conclusion Mass-action Kinetics Assumption: number of effective reactant collisions Z proportional to reactant concentrations (Guldberg 1867) ˆ k A � B Ý Ñ C . . . Z C ✒ r A s and Z C ✒ r B s ñ Z C ✒ r A s ☎ r B s Production rate generating C: V prod ♣r C sq ✏ ˆ k ☎ r A s ☎ r B s Chemical Analog Computing with Negative Numbers Hendrik Happe 4/ 30
Introduction Mass-action Kinetics Negative Numbers Conclusion Mass-action Kinetics Assumption: number of effective reactant collisions Z proportional to reactant concentrations (Guldberg 1867) ˆ k A � B Ý Ñ C . . . Z C ✒ r A s and Z C ✒ r B s ñ Z C ✒ r A s ☎ r B s Production rate generating C: V prod ♣r C sq ✏ ˆ k ☎ r A s ☎ r B s Consumption rate of C V cons ♣r C sq ✏ 0 d C d t ✏ V prod ♣r C sq ✁ V cons ♣r C sq ✏ ˆ k ☎ r A s ☎ r B s Chemical Analog Computing with Negative Numbers Hendrik Happe 4/ 30
� � � Ý Ñ � � � � � Ý Ñ � � � � � Ý Ñ � � ✄ ☛ r s ✏ ✾ ➳ ➵ r s ✏ ☎ ♣ ✁ q ☎ r s ✏ ✏ ✏ Introduction Mass-action Kinetics Negative Numbers Conclusion Mass-action Kinetics: General ODE Model Considering scheme of r reaction and p involved species Chemical Analog Computing with Negative Numbers Hendrik Happe 5/ 30
✄ ☛ r s ✏ ✾ ➳ ➵ r s ✏ ☎ ♣ ✁ q ☎ r s ✏ ✏ ✏ Introduction Mass-action Kinetics Negative Numbers Conclusion Mass-action Kinetics: General ODE Model Considering scheme of r reaction and p involved species k 1 a 1 , 1 S 1 � a 2 , 1 S 2 � . . . � a p , 1 S p Ý Ñ b 1 , 1 S 1 � b 2 , 1 S 2 . . . � b p , 1 S p k 2 a 1 , 2 S 1 � a 2 , 2 S 2 � . . . � a p , 2 S p Ý Ñ b 1 , 2 S 1 � b 2 , 2 S 2 . . . � b p , 2 S p . . . k r a 1 , r S 1 � a 2 , r S 2 � . . . � a p , r S p Ý Ñ b 1 , r S 1 � b 2 , r S 2 . . . � b p , r S p Results in system of ordinary differential equations (ODEs): Chemical Analog Computing with Negative Numbers Hendrik Happe 5/ 30
Introduction Mass-action Kinetics Negative Numbers Conclusion Mass-action Kinetics: General ODE Model Considering scheme of r reaction and p involved species k 1 a 1 , 1 S 1 � a 2 , 1 S 2 � . . . � a p , 1 S p Ý Ñ b 1 , 1 S 1 � b 2 , 1 S 2 . . . � b p , 1 S p k 2 a 1 , 2 S 1 � a 2 , 2 S 2 � . . . � a p , 2 S p Ý Ñ b 1 , 2 S 1 � b 2 , 2 S 2 . . . � b p , 2 S p . . . k r a 1 , r S 1 � a 2 , r S 2 � . . . � a p , r S p Ý Ñ b 1 , r S 1 � b 2 , r S 2 . . . � b p , r S p Results in system of ordinary differential equations (ODEs): ✄ p ☛ r r S i s ✏ d r S i s ✾ ➳ ➵ r S j s a j , h ✏ k h ☎ ♣ b i , h ✁ a i , h q ☎ with i ✏ 1 , . . . , p d t h ✏ 1 j ✏ 1 Chemical Analog Computing with Negative Numbers Hendrik Happe 5/ 30
ñ r s ✏r s � r s ✾ r s ✏ ✾ r s ✏ ✾ r s ✏ r s � r s ✁ r s ✏ ✏ ✁ r s♣✽q ✏ Ñ✽ ♣ ✁ q ☎ ♣r s♣ q � r s♣ qq ✏ r s♣ q � r s♣ q Introduction Mass-action Kinetics Negative Numbers Conclusion Chemical Calculations: Addition X 1 k 1 k 3 Y ❍ X 2 k 2 Chemical Analog Computing with Negative Numbers Hendrik Happe 6/ 30
ñ r s ✏r s � r s ✏ ✏ ✁ r s♣✽q ✏ Ñ✽ ♣ ✁ q ☎ ♣r s♣ q � r s♣ qq ✏ r s♣ q � r s♣ q Introduction Mass-action Kinetics Negative Numbers Conclusion Chemical Calculations: Addition X 1 k 1 ✾ r X 1 s ✏ 0 k 3 ✾ Y ❍ r X 2 s ✏ 0 ✾ X 2 k 2 r Y s ✏ k 1 r X 1 s � k 2 r X 2 s ✁ k 3 r Y s Chemical Analog Computing with Negative Numbers Hendrik Happe 6/ 30
ñ r s ✏r s � r s Introduction Mass-action Kinetics Negative Numbers Conclusion Chemical Calculations: Addition X 1 k 1 ✾ r X 1 s ✏ 0 k 3 ✾ Y ❍ r X 2 s ✏ 0 ✾ X 2 k 2 r Y s ✏ k 1 r X 1 s � k 2 r X 2 s ✁ k 3 r Y s asymptotic solution in a stationary state with k 1 ✏ k 2 ✏ k 3 t Ñ✽ ♣ 1 ✁ e ✁ k 1 t q ☎ ♣r X 1 s♣ t q � r X 2 s♣ t qq ✏ r X 1 s♣ 0 q � r X 2 s♣ 0 q r Y s♣✽q ✏ lim Chemical Analog Computing with Negative Numbers Hendrik Happe 6/ 30
Introduction Mass-action Kinetics Negative Numbers Conclusion Chemical Calculations: Addition X 1 k 1 ✾ r X 1 s ✏ 0 k 3 ✾ Y ❍ r X 2 s ✏ 0 ✾ X 2 k 2 r Y s ✏ k 1 r X 1 s � k 2 r X 2 s ✁ k 3 r Y s asymptotic solution in a stationary state with k 1 ✏ k 2 ✏ k 3 t Ñ✽ ♣ 1 ✁ e ✁ k 1 t q ☎ ♣r X 1 s♣ t q � r X 2 s♣ t qq ✏ r X 1 s♣ 0 q � r X 2 s♣ 0 q r Y s♣✽q ✏ lim ñ r Y s ✏r X 1 s � r X 2 s Chemical Analog Computing with Negative Numbers Hendrik Happe 6/ 30
s ✾ ñ r s ✏r ✁r s ✾ s ✏ ✾ r r s ✏ ✾ r s ✏ ✁ r sr s ✁ r s � r s ✾ r s ✏ r s ✁ r sr s ✏ ★ r s♣ q ✁ r s♣ q r s♣ q → r s♣ q r s♣✽q ✏ Introduction Mass-action Kinetics Negative Numbers Conclusion Chemical Calculations: Non-negative Subtraction X 2 k 1 k 2 ❍ Z k 1 ❍ Y X 1 k 1 Chemical Analog Computing with Negative Numbers Hendrik Happe 7/ 30
s ✾ ñ r s ✏r ✁r s ✏ ★ r s♣ q ✁ r s♣ q r s♣ q → r s♣ q r s♣✽q ✏ Introduction Mass-action Kinetics Negative Numbers Conclusion Chemical Calculations: Non-negative Subtraction X 2 k 1 r X 1 s ✏ ✾ ✾ r X 2 s ✏ 0 k 2 ✾ r Y s ✏ ✁ k 2 r Y sr Z s ✁ k 1 r Y s � k 1 r X 1 s ❍ Z ✾ r Z s ✏ k 1 r X 2 s ✁ k 2 r Y sr Z s k 1 ❍ Y X 1 k 1 Chemical Analog Computing with Negative Numbers Hendrik Happe 7/ 30
s ✾ ñ r s ✏r ✁r s Introduction Mass-action Kinetics Negative Numbers Conclusion Chemical Calculations: Non-negative Subtraction X 2 k 1 r X 1 s ✏ ✾ ✾ r X 2 s ✏ 0 k 2 ✾ r Y s ✏ ✁ k 2 r Y sr Z s ✁ k 1 r Y s � k 1 r X 1 s ❍ Z ✾ r Z s ✏ k 1 r X 2 s ✁ k 2 r Y sr Z s k 1 ❍ Y X 1 k 1 asymptotic solution in a stationary state with k 1 ✏ k 2 ★ r X 1 s♣ 0 q ✁ r X 2 s♣ 0 q if r X 1 s♣ 0 q → r X 2 s♣ 0 q r Y s♣✽q ✏ 0 else Chemical Analog Computing with Negative Numbers Hendrik Happe 7/ 30
Introduction Mass-action Kinetics Negative Numbers Conclusion Chemical Calculations: Non-negative Subtraction X 2 k 1 r X 1 s ✏ ✾ ✾ r X 2 s ✏ 0 k 2 ✾ r Y s ✏ ✁ k 2 r Y sr Z s ✁ k 1 r Y s � k 1 r X 1 s ❍ Z ✾ r Z s ✏ k 1 r X 2 s ✁ k 2 r Y sr Z s k 1 ❍ Y X 1 k 1 asymptotic solution in a stationary state with k 1 ✏ k 2 ★ r X 1 s♣ 0 q ✁ r X 2 s♣ 0 q if r X 1 s♣ 0 q → r X 2 s♣ 0 q r Y s♣✽q ✏ 0 else ✏r X 1 s ✾ ñ r Y s ✁r X 2 s Chemical Analog Computing with Negative Numbers Hendrik Happe 7/ 30
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