35. FUNICULÌ, FUNICULÀ Field-drive is a locomotion mechanism that is analogous to general relativis- tic warp-drive. In this mechanism, an active particle continuously climbs up the field-gradient generated by its own influence on the environment so that the particle can bootstrap itself into a constant non-zero velocity motion. Consider a field-drive in one-dimensional (the Or axis) environment, where the position of the particle at time t is given by X(t) and its instantaneous velocity follows from: d dt X(t) = K Ә əx R(x, t) x=X(t) in which is called the guiding coefficient and R(x, t) is the field-value in this space. Note that, the operation.... y means you have to calculate the part in first, then replace with X(t) For a
35. FUNICULÌ, FUNICULÀ Field-drive is a locomotion mechanism that is analogous to general relativis- tic warp-drive. In this mechanism, an active particle continuously climbs up the field-gradient generated by its own influence on the environment so that the particle can bootstrap itself into a constant non-zero velocity motion. Consider a field-drive in one-dimensional (the Or axis) environment, where the position of the particle at time t is given by X(t) and its instantaneous velocity follows from: d dt X(t) = K Ә əx R(x, t) x=X(t) in which is called the guiding coefficient and R(x, t) is the field-value in this space. Note that, the operation.... y means you have to calculate the part in first, then replace with X(t) For a
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Transcribed Image Text:35. FUNICULI, FUNICULA Field-drive is a locomotion mechanism that is analogous to general relativis-
tic warp-drive. In this mechanism, an active particle continuously climbs up the field-gradient generated
by its own influence on the environment so that the particle can bootstrap itself into a constant non-zero
velocity motion. Consider a field-drive in one-dimensional (the Ox axis) environment, where the position
of the particle at time t is given by X(t) and its instantaneous velocity follows from:
d
dt X(t) = k
ə
Ət
-R(x, t)
x=X(t)
in which is called the guiding coefficient and R(x, t) is the field-value in this space. Note that, the
operation ...z-X(t) means you have to calculate the part in ... first, then replace x with X(t). For a
biological example, the active particle can be a cell, the field can be the nutrient concentration, and the
strategy of climbing up the gradient can be chemotaxis. The cell consumes the nutrient and also responses
to the local nutrient concentration, biasing its movement toward the direction where the concentration
increases the most. If the nutrient is not diffusive and always recovers locally (e.g. a surface secretion) to
the value which we defined to be 0, then its dynamics can usually be approximated by:
Ә
əx
1
R(x,t) = — -R(x, t) — y exp
TRAVELLED PATH
[x - X(t)1²)
XOP}
where is the timescale of recovery, y is the consumption, and A is the characteristic radius of influence.
Before we inoculate the cell into the environment, R = 0 everywhere at any time. What is the smallest
guiding coefficient (in um²/s) for field-drive to emerge, if the parameters are 7 = 50s, y = 1s-¹, and
A = 10μm.
21²
ACTIVE
PARTICLE
FIELD
DRIVE
NON-ZERO
GRADIENT
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