Principles of Electronic Nanobiosensors
Unit 2: Settling Time
Lecture 2.8: First Passage and Narrow Escape Time I
By Muhammad A. Alam
Professor of Electrical and Computer Engineering Purdue University
alam@purdue.edu
1
Outline
•‘Blind’ molecule escaping from a cell
•First passage time (FPT)
•First passage time (FPT): examples
•Narrow escape time (NET)
•Conclusion
2
On Cells and Sensors
|
|
|
|
|
|
lysosome |
|
|
|
|
|
|
|
|
|
|
|
|
|
||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||||
|
|
|
|
|
centriole |
|
ribosomes |
|
|
cilium |
|
|
|
|
|||||||
|
|
centrosome |
|
|
|
|
|
|
|
|
|
|
Cell |
|
|
|
|||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||
|
|
|
|
|
|
|
|
|
|
|
membrane |
|
|
|
|||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||
|
peroxisome |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||
|
|
|
|
|
|
|
|
|
|
|
|
|
Smooth |
|
|||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
endoplasmic |
|
||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
reticulum |
|
||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Nuclear pore |
|
||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Nucleolus |
|
||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Nucleoplasm |
|
||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Nuclear |
|
||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
envelop |
|
||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Rough |
|
||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
endoplasmic |
|
||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
reticulum |
|
||
Golgi |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||||
apparatus |
|
|
|
|
|
|
|
|
mitochondrion |
|
|
||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Secretory |
|
cytoplasm |
|
|
|
|
|
|
|
|
|
|
||||
|
|
|
|
|
vesicles |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Target
nucleus
I2
X0
Q = ∫ρdυ
V
|
|
|
|
|
I1 |
|
|
3 |
|
|
||||
Settling time, MFPT, and NET
Settling time
First passage time (FPT) |
Narrow escape time (NET) |
4