Dilute bacterial suspensions 18.S995 - L06 & 07 dunkel@mit.edu - PowerPoint PPT Presentation

Dilute bacterial suspensions 18.S995 - L06 & 07 dunkel@mit.edu

E.coli (non-tumbling HCB 437) Drescher, Dunkel, Ganguly, Cisneros, Goldstein (2011) PNAS dunkel@math.mit.edu

E.coli (non-tumbling HCB 437) eed V 0 = 22 ± 5 µ m/s. A A = � F r = r h i r . ˆ d ) 2 − 1 | r | , th � = 1 . 9 µ m ˆ u ( r ) = 3( ˆ r , ˆ 8 πη , regions, we obtai | r | 2 rce F = 0 . 42 pN. Drescher, Dunkel, Ganguly, Cisneros, Goldstein (2011) PNAS dunkel@math.mit.edu

Bacterial run & tumble motion ~20 parts V. Kantsler movie: 20 nm source: wiki Berg (1999) Physics Today Chen et al (2011) EMBO Journal

Bacterial run & tumble motion V. Kantsler movie: Rep. Prog. Phys. 72 (2009) 096601 E Lauga and T R Powers (a) (b) (c) (d) Figure 15. Bundling of bacterial flagella. During swimming, the bacterial flagella are gathered in a tight bundle behind the cell as it moves through the fluid (( a ) and ( d )). During a tumbling event, the flagella come out the bundle ( b ), resulting in a random reorientation of the cell before the next swimming event. At the conclusion of the tumbling event, hydrodynamic interactions lead to the relative attraction of the flagella ( c ), and their synchronization to form a perfect bundle ( d ).

for more movies, see also http://www.rowland.harvard.edu/labs/bacteria/movies/index.php dunkel@math.mit.edu

week ending P H Y S I C A L R E V I E W L E T T E R S PRL 101, 038102 (2008) 18 JULY 2008 Hydrodynamic Attraction of Swimming Microorganisms by Surfaces Allison P. Berke, 1 Linda Turner, 2 Howard C. Berg, 2,3 and Eric Lauga 4, * 1 Department of Mathematics, Massachusetts Institute of Technology, 77 Mass. Avenue, Cambridge, Massachusetts 02139, USA 60 120 H = 100 µ m (a) n ( y ) [2N cells] n ( y ) [N cells] exp . ( N cells) theory exp . (2 N cells) 80 40 theory H (b) 20 40 0 0 0 20 40 60 80 100 y ( µ m ) y (c) 60 H = 200 µ m exp . theory 40 n ( y ) 20 0 0 50 100 150 200 y ( µ m ) (d)

Goals • minimal SDE model for microbial swimming • wall accumulation & density profile dunkel@math.mit.edu

1.3 Dilute microbial suspensions A minimalist model for the locomotion of an isolated microorganism (e.g., alga or bac- terium) with position X ( t ) and orientation unit vector N ( t ) is given by the coupled system of Ito SDEs p = V N dt + 2 D T ⇤ d B ( t ) , (1.45a) d X p = (1 � d ) D R N dt + 2 D R ( I � NN ) ⇤ d W ( t ) . (1.45b) d N To confirm that Eq. (1.45b) conserves the unit length of the orientation vector, | N | 2 = 1 for all t , it is convenient to rewrite Eqs. (1.45) in component form: p = V N i dt + 2 D T ⇤ dB i ( t ) , (1.46a) dX i p = (1 � d ) D R N j dt + 2 D R ( δ jk � N j N k ) ⇤ dW k ( t ) . (1.46b) dN j http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.0020044

BM on the unit sphere p ⇤ p = (1 � d ) D R N j dt + 2 D R ( δ jk � N j N k ) ⇤ dW k ( t ) . dN j p � � ⇤ For the constraint | N | 2 = 1 to be satisfied, we must have d | N | 2 = 0. Applying the d - dimensional version of Ito’s formula, see Eq. (A.12), to F ( N ) = | N | 2 , one finds indeed that d | N | 2 � � = 2 N j ⇤ dN j + D R ( δ ij � N i N j ) dt ∂ N i ∂ N j N k N k h i p = 2 N j ⇤ (1 � d ) D R N j dt + 2 D R ( δ jk � N j N k ) ⇤ dW k ( t ) + ∂ N i ( δ jk N k + N k δ jk ) D R ( δ ij � N i N j ) dt = 2(1 � d ) D R dt + ( δ jk δ ik + δ ik δ jk ) D R ( δ ij � N i N j ) dt = 0 . (1.47)

Orientation correlations To understand the dynamics (1.46), it is useful to compute the orientation correlation, h N ( t ) · N (0) i = E [ N ( t ) · N (0)] = E [ N z ( t )] , (1.48) where we have assumed (w.l.o.g.) that N (0) = e z . Averaging Eq. (1.46b), we find that d dt E [ N z ( t )] = (1 � d ) D R E [ N z ( t )] , (1.49) implying that, in this model, the memory loss about the orientation is exponential h N ( t ) · N (0) i = e (1 − d ) D R t , (1.50)

Mean square displacement d | X | 2 � � = 2 X j ⇤ dX j + ∂ X i ∂ X j X k X k D T δ ij dt = 2 X j ⇤ dX j + ( δ jk δ ik + δ ik δ jk ) D T δ ij dt = 2 X j ⇤ dX j + 2 d D T dt p = 2 X j [ V N j dt + 2 D T ⇤ dB j ( t )] + 2 d D T dt, (1.51) averaging and dividing by dt , gives d dt E [ X 2 ] = 2 V E [ X ( t ) N ( t )] + 2 d D T . (1.52) The expectation value on the rhs. can be evaluated by making use of Eq. (1.50): Z t � E [ X ( t ) · N ( t )] = d X ( s ) · N ( t ) E 0 Z t � = ds N ( s ) · N ( t ) V E 0 Z t = ds h N ( t ) · N ( s ) i V 0 Z t ds e (1 − d ) D R ( t − s ) = V 0 V 1 � e (1 − d ) D R t ⇤ ⇥ = . ( d � 1) D R

Mean square displacement ⇥ ⇤ � By inserting this expression into Eq. (1.52) and integrating over t , we find 2 V 2 ( d � 1) D R t + e (1 − d ) D R t � 1 E [ X 2 ] = ⇥ ⇤ + 2 dD T t. (1.53) ( d � 1) 2 D 2 R If D T is small, then at short times t ⌧ D − 1 R the motion is ballistic E [ X 2 ] ' V 2 t 2 + 2 dD T t, (1.54) At large times, the motion becomes di ff usive, with asymptotic di ff usion constant E [ X 2 ] 2 V 2 lim = + 2 dD T . (1.55) ( d � 1) D R t t →∞ Inserting typical values for bacteria, V ⇠ 10 µ m/s and D R ⇠ 0 . 1/s, and comparing with D T ⇠ 0 . 2 µ m 2 /s for a micron-sized colloids at room temperature, we see that active swim- ming and orientational di ff usion dominate the di ff usive dynamics of microorganisms at long times.

week ending P H Y S I C A L R E V I E W L E T T E R S PRL 101, 038102 (2008) 18 JULY 2008 Hydrodynamic Attraction of Swimming Microorganisms by Surfaces Allison P. Berke, 1 Linda Turner, 2 Howard C. Berg, 2,3 and Eric Lauga 4, * 1 Department of Mathematics, Massachusetts Institute of Technology, 77 Mass. Avenue, Cambridge, Massachusetts 02139, USA 60 120 H = 100 µ m (a) n ( y ) [2N cells] n ( y ) [N cells] exp . ( N cells) theory exp . (2 N cells) 80 40 theory H (b) 20 40 0 0 0 20 40 60 80 100 y ( µ m ) y (c) 60 H = 200 µ m exp . theory 40 n ( y ) 20 0 0 50 100 150 200 y ( µ m ) (d)

‘Hydrodynamic’ fields An interesting question that is relevant Concentration profile between two walls from a medical perspective concerns the spatial distribution of bacteria and other swimming microbes in the presence of confinement. Restricting ourselves to dilute suspensions 10 , we may obtain a simple prediction from the model (1.45) by considering the FPE for the associated PDF p ( t, x , n ). Given p and the total number of bacteria N b in the solutions, we obtain the spatial concentration profile by integrating over all possible orientations Z c ( t, x ) = N b d n p ( t, n , x ) . (1.56a) S d The associated mean orientation field reads Z u ( t, x ) = N b d n p ( t, n , x ) n . (1.56b) S d The FPE for the Ito-SDE (1.45) can be written as a conservation law = � ( ∂ x i J i + ∂ n i Ω i ) , (1.57a) ∂ t p where = ( V n i � D T ∂ x i ) p (1.57b) J i � = (1 � d ) n i p � ∂ n j [( δ ij � n i n j ) p ] (1.57c) Ω i D R .

Concentration field Z The FPE for the Ito-SDE (1.45) can be written as a conservation law ∂ t p = � ( ∂ x i J i + ∂ n i Ω i ) , (1.57a) where = ( V n i � D T ∂ x i ) p (1.57b) J i � = (1 � d ) n i p � ∂ n j [( δ ij � n i n j ) p ] (1.57c) Ω i D R . Focusing on the three-dimensional case, d = 3, we are interested in deriving from Eq. (1.57) the stationary concentration profile c of a suspension that is confined by two quasi-infinite parallel walls, which are located z = ± H . That is, we assume that the distance between the walls is much smaller then their spatial extent in the ( x, y )-directions, 2 H ⌧ L x , L y . To obtain an evolution equation for c , we multiply Eq. (1.57a) by N b and integrate over n with Z d n ∂ n i Ω i = 0 . (1.58) S d This yields the mass conservation law = �r · ( V u � D T r c ) . (1.59) ∂ t c

Orientation (velocity) field Z The FPE for the Ito-SDE (1.45) can be written as a conservation law = � ( ∂ x i J i + ∂ n i Ω i ) , (1.57a) ∂ t p where = ( V n i � D T ∂ x i ) p (1.57b) J i � = (1 � d ) n i p � ∂ n j [( δ ij � n i n j ) p ] (1.57c) Ω i D R . �r · � r To obtain also an evolution equation for u , we multiply Eq. (1.57a) by n k , ∂ t ( n k p ) = � ∂ x i ( n k J i ) � n k ∂ n i Ω i . (1.60) and note that n k ∂ n i Ω i = ∂ n i ( n k Ω i ) � ( ∂ n i n k ) Ω i = ∂ n i ( n k Ω i ) � δ ik Ω i . (1.61)